DISCLOSURE / FILE
An 11 March 2010 AAWSAP Defense Intelligence Reference Document that rewrites the 1961 Drake Equation as a product of seven random variables, shows through the Central Limit Theorem that the number of communicating civilizations N follows a lognormal distribution, and derives a probability distribution for the distance to the nearest one.
DOW-UAP-D127, AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010
An 11 March 2010 AAWSAP Defense Intelligence Reference Document that rewrites the 1961 Drake Equation as a product of seven random variables, shows through the Central Limit Theorem that the number of communicating civilizations N follows a lognormal distribution, and derives a probability distribution for the distance to the nearest one.
Dated 11 March 2010 and marked UNCLASSIFIED//FOR OFFICIAL USE ONLY, the document was prepared by the Acquisition Support Division (DWO-3) of the DIA Defense Warning Office as one of the FY 2009 Advanced Aerospace Weapon System Applications (AAWSA) reports; the author line is redacted as 'AAP Person 80,' but the body identifies the author as the writer of paper IAC-08-A4.1.4, which Appendix B reproduces in full under Claudio Maccone's name. The method replaces each Drake factor with a uniformly distributed random variable (justified by Shannon's 1948 maximum-entropy result), takes natural logs to turn the product into a sum, and applies the Central Limit Theorem so that N tends to a lognormal distribution. For one illustrative input set the classical product gives N = 3500, while the statistical version gives a mode near 250, a median near 1740, a mean of about 4590 and a standard deviation of 11,195. The same machinery yields a distance distribution with a mode of 1933 light years, a mean of 2670, a standard deviation of 1309 and a 75 percent band between 1361 and 3979 light years; the document states plainly that its inputs are assumed for illustration and that it makes no assumption about the astronomy, biology or sociology behind them.
This product is one in a series of advanced technology reports produced in FY 2009 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapon System Applications (AAWSA) Program.p.2
This is a crazy assumption, clearly, and should be replaced by more scientifically-grounded assumptions as soon as we know more about our Galactic Neighborhood.p.5
In all cases, however, the final result about N has always been a sheer number, i.e., a positive integer number ranging from 1 to millions or billions.p.11
Thus, N tends to the LOGNORMAL DISTRIBUTION.p.13
We see that, for this particular set of seven inputs, the classical Drake equation (i.e. the product of the seven numbers) yields a total of 3500 communicating extraterrestrial civilizations existing in the galaxy right now.p.16
In plain words: with 75 percent probability, the nearest extraterrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by table 1.p.23
So, we really make no assumption about the astronomy, or the biology, or the sociology of the Drake equation: we just care about its mathematics.p.34
My apologies, Frank, for disrupting the beautiful simplicity of your equationp.54
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UNCLASSIFIED/ /POil OFFl@IAI:: WSli 0Nk¥ Defense Intelligence Reference Document Acquisition Threat Support 11 March 20 10 !COD : 1 December 2009 An Introduction to the Statistical Drake Equation UNCLASSIFIED/ /FOA OFFICIO L: 1 PiF ON! Y
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UNCLASSIFIED//P81t 8FFIEIAL Y&E &ttbl/ An Introduction to the Statistical Drake Equation Prepared by: Acquisition Support Division (DW0-3) Defense Warning Office Directorate for Analysis Defense Intelligence Agency Author: AAP Person 80 Administrative Note COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized . This product is one in a series of advanced technology reports produced in FY 2009 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapon System Applications (AAWSA) Program. Comments or questions pertaining to this document should be addressed t o !AAP Person 1 ~ AAWSA Prog ram Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington, DC 20340-5100. UNCLASSIFIED//F8R 8FFl61tlib Wlilli Qtlb¥ ii
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UNCLASSIFIED/ /FOR OFFIEil.t.k W&i Q~lls¥ References [1] Benford, Gregory, Jim and Dominic, "Cost Optimized Interstellar Beacons: SETI", arXiv.org web site (22 Oct. 2008). [2] Carl Sagan, "Cosmos", Random House, New York, 1983. See in particular the pages 298-302. [3] Bennet, Jeffrey, and Shostak, Seth, "Life in the Universe", second edition, Pearson - Addison Wesley, San Francisco, 2007. See in particular page 404. [4] C. Maccone, "The Statistical Drake Equation", paper #IAC-08-A4.1.4 presented on October 1st, 2008, at the 59 th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29 th thru October 3rd, 2008. UNCLASSIFIED/ /FOR OFFIEil.t.k W&i QNls¥ 55
Concatenated page-by-page transcript. Born-digital pages came through pdf.js; scanned pages were transcribed by Claude vision OCR. Pages marked unreadable failed multiple OCR retries (heavy redaction, microfilm artifacts, or blank separators) and are kept in place for audit.
UNCLASSIFIED/ /POil OFFl@IAI:: WSli 0Nk¥ Defense Intelligence Reference Document Acquisition Threat Support 11 March 20 10 !COD : 1 December 2009 An Introduction to the Statistical Drake Equation UNCLASSIFIED/ /FOA OFFICIO L: 1 PiF ON! Y
UNCLASSIFIED//P81t 8FFIEIAL Y&E &ttbl/ An Introduction to the Statistical Drake Equation Prepared by: Acquisition Support Division (DW0-3) Defense Warning Office Directorate for Analysis Defense Intelligence Agency Author: AAP Person 80 Administrative Note COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized . This product is one in a series of advanced technology reports produced in FY 2009 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapon System Applications (AAWSA) Program. Comments or questions pertaining to this document should be addressed t o !AAP Person 1 ~ AAWSA Prog ram Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington, DC 20340-5100. UNCLASSIFIED//F8R 8FFl61tlib Wlilli Qtlb¥ ii
UNCLASSIFIED/ /POil OFFl@IAL WSli 0Nk¥ Contents 1. Introduction ....................................................................................................... iv 2. The Key Question: How Far are They ? .............................................................. 4 3. Computing N By Virtue of the Drake Equation (1961) ........................................ 7 4. The Drake Equation is Over-Simplified ............................................................. 10 5. The Statistical Drake Equation ......................................................................... 11 6. Solving the Statistical Drake Equation By Virtue of the Central Limit Theorem (CLT) of Statistics .......................................................... ,.................................... 13 7. An Example Explaining the Statistical Drake Equation ..................................... 15 8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical Drake Equation ................................................................................................. 18 9. The "Data Enrichment Principle" as the Best CLT Consequence Upon the Statistical Drake Equation (Any Number of Factors Allowed) ........................... 23 10. Conclusions .................................................................................................... 23 Appendix A: Proof of Shannon's 1948 Theorem Stating That the Uniform Distribution is the "Most Uncertain" One Over a Finite Range of Appendix B: Original Text of the Author's Paper #IAC-08-A4.1.4 Entitled the Values ............................................................................................. 25 Statistical Drake Equation ............................................................... 28 References ........................................................................................................... 55 UNCLASSIFIED/ fEOR OFFICIO! 11SF ON! X iii
UNCLASSIFIED/ /POil OFFl@IAI:: WSli 0Nk¥ An Introduction to the Statistical Drake Equation 1. Introduction SETI (an acronym for "Search for Extraterrestrial Intelligence") is a relatively new branch of scientific research, having begun only in 1959. Its goal is to ascertain whether alien civilizations exist in the universe, how far from us they exist, and possibly how much more advanced than us they may be. As of 2009, the only physical tools we know that could help us get in touch with aliens are the electromagnetic waves an alien civilization could emit and we could detect. This forces us to use the largest radiotelescopes on Earth for SETI research, because the higher our collecting area of electromagnetic radiation is, the higher our sensitivity is (that is, the farther in space we can probe). Yet, even by using the largest radiotelescopes on Earth (the 310-meter dish at Arecibo, for instance), we cannot search for aliens beyond, say, a few hundred light years away. This is a very, very small amount of space around us within our galaxy, the Milky Way, that is about 100,000 light years in diameter. Thus, current SETI can cover only a very tiny fraction of the galaxy, and it is not surprising that in the past 50 years of SETI searches, NO extraterrestrial civilization was discovered. Quite simply, we did not get far enough! This demands the construction of much more powerful and radically new radiotelescopes. Rather than big and heavy metal dishes, whose mechanical problems hamper SETI research too much, we are now turning to "software radiotelescopes," where a large number of small dishes (ATA = Allen Telescope Array, and ALMA= Atacama Large Millimeter/submillimeter Array) or even just of simple dipoles (LOFAR = Low Frequency Array) using state-of the-art electronics and very- high-speed computing can outperform the classical radiotelescopes in many regards. The final dream in this field is the SKA ( = Square Kilometer Array), currently being designed and expected to be completed around 2020. 2. The Key Question: How Far are They? But still, the key question remains: how far are they? Or, more correctly, how far do we expect the NEAREST extraterrestrial civilization to be from t he Solar System in the galaxy? This question was first faced in a scientific manner back in 1961 by the same scientist who also was t he first experimental SETI rad io astronomer ever: t he American, Fra nk Donald Drake (born 1930). He first considered the shape and size of the galaxy where we are living: the Milky Way. This is a spiral ga laxy measuring some 100,000 light years in diameter and some 16,000 light years in thickness of the Ga lactic Disk at half way from its center. That is: The diameter of the galaxy is (about) 100,000 light years, (abbreviated ly) i.e., its radius, R Gala.1y , is about 50,000 ly. UNCLA SSIFIED/ /FOR OFFI&IJ.k Wili Ql'II.¥ iv
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ The thickness of the Galactic Disk at half-way from its center, h a " 1t'-'Y ' is about 16,000 ly. The volume of the galaxy may then be approximated as the volume of the corresponding cylinder, i.e. (1) Now consider the sphere around us having a radius r. The volume of such a sphere is 4 ( Ef Distance ) 3 V o ur _ Sphere = 3 n - (2) 2 In the last equation, we had to divide the distance "ET_Distance" between ourselves and the nearest ET civilization by 2 because we are now going to make the unwarranted assumption that all ET civilizations are equally spaced from each other in the galaxy! This is a crazy assumption, clearly, and should be replaced by more scientifically-grounded assumptions as soon as we know more about our Galactic Neighborhood. At the moment, however, this is the best guess that we can make, and so we shall take it for granted, although we are aware that this is a weak point in the reasoning. Furthermore, let us denote by N the total number of civilizations now living in the galaxy, including ourselves. Of course, this number N is unknown. We only know that N ~ 1 since one civilization does at least exist! Having thus assumed that ET civilizations are UNIFORMLY SPACED IN THE GALAXY, we can then write down the proportion: V a a/a.,y V o ,, r _ Spher e -- =-~- (3) N That is, upon replacing both (1) and (2) into (3): 3 4 ( Ef - Dis lance ) 2 - i'l" _ ___ i'l" R Ga /axy h = 3 2 (4) N 1 The last equation contains two unknowns: N and ET_Distance, and so we don't know which one it is better to solve for. However, we may suppose that, by resorting to the (rather uncertain) knowledge that we have about the Evolution of the galaxy through the last 10 billion years or so, we might somehow compute an approximate value for N. Then, we may solve (4) for ET_Distance thus obtaining the (AVERAGE) DISTANCE BETWEEN ANY PAIR OF NEIGHBORING CIVILIZATIONS IN THE GALAXY (DISTANCE LAW) UNCLASSIFIED/ ; 'FQA QFFICiIPL. !Iii 011! Y 5
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ 3 6 R 2 h Ef ff (N) Ca!a,y _ ,stance = VN C (5) VN where the posit ive constant C is defined by C = V6 R ~ala.,y h ca /axy ,., 28845 light years . (6) Equations (5) and (6) are the starting point to understand t he orig in of the Drake equation that we discuss in detail in Section 3 of th is paper. Let us just complete this section by pointing out three different numerical cases of the distance law (5): • We know that we exist, so N may not be smaller tha n 1, i.e., N ~ 1. Suppose then that we are alone in the galaxy, i.e., that N=l. Then the distance law (5) yields as distance to the nearest civilization from us just the constant C, i.e., 28,845 light years. Th is is about the distance in between ourselves and the center of the galaxy (i. e. the Galactic Bulge) . Thus, this result seems to suggest that, if we do not find any extraterrestrial civilization around us in these outskirts of the galaxy where we live, we should look around the Galactic Center first. And this is indeed what is happening, i.e., many SETI searches are actually point ing the antennas towards the Galactic Center, looking for beacons (see, for instance ref. [1]). • Suppose next that N=l000, i.e. there are about a thousand extraterrestrial communicating civilizations in the whole galaxy right now. Then the distance law (5) yields an average distance of 2,885 light yea rs. This is a distance that most radiotelescopes in Earth may not reach for SETI searches right now: hence the need to build larger radiotelescopes, like ALMA, LOFAR and the SKA. • Suppose finally that N=l00000O, i.e., there are a million communicating civilizations now in the galaxy. Then the distance law (5) yields an average dista nce of 288 light years. Th is is with in the (upper) range of distances that our current rad iotelescopes may reach for SETI searches, and that justifies all SETI searches that have been done so far in t he first fifty years of SETI (1960-2010). In conclusion, interpolating the above three special cases of N, we may say that the distance law (5) yields t he following key diagram of the average ET distance vs. the assumed number of communicating civilizations, N, in the galaxy right now (Figure 1): UNCLASSIFIED/ J 'FOA. OFlilCI0 I. P !SE ON! X 6
0 l \ ' "- - ~ UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ Av era ge DIST A CE o f the nea res t ET c iviliza tion vs . th e ASSUM ED NUMBffi of ET c ivilizations in th e Gah Cl) 0::: <( UJ > ! :I: :i .!:: !'.l g ~ !'l = 200 175 150 125 100 75 0 500 25 0 0 0 I 00000 200000 300000 400000 500000 600000 700000 800000 900000 I000000 ASS MED NUMB ER of civiliz ations in the Galaxy (that is, Nin the Drake equation) Figure 1. DISTANCE LAW; i.e., the Average Distance (plot along the ver-1:ical axis in light years) Versus the NUMBER of Communicating Civilizations ASSUMED to Exist in the Galaxy Right Now 3. Computing N By Virtue of the Drake Equation (1961) In the previous section, the problem of finding how close the nearest ET civilization may be was "solved" by reducing it to the computation of N, the total number of extraterrestrial civilizations now existing in this galaxy. In this section the famous Drake equation is described, that was proposed back in 1961 by Frank Dona ld Drake (born 1930) to estimate the numerical value of N. We believe that no better introductory description of the Drake equations exists other than the one given by Carl Sagan in his 1983 book "Cosmos" (ref. [2]), in its turn based on the famous TV series "Cosmos." So, in this paragraph we report Carl Sagan's description of the Drake equation unabridged. "But is there anyone out there to talk to? With a third or a half a trillion stars in our Milky Way galaxy alone, could ours be the only one accompanied by an inhabited planet? How much more likely it is that technical civilizations are a cosm ic commonplace, that the galaxy is pulsing and humming with advanced societies, and, therefore, that the nearest such culture is not so very far away - perhaps transmitting from antennas established on a planet of a naked-eye star just next door. Perhaps when we look up at the sky at nig ht, near one of those faint pinpoints of light is a world on which someone quite different from us is then glancing id ly at a star we call the Sun and entertaining, for just a moment, an outrageous speculation. UNCLASSIFIED//FOR OFFIEiIAk Wlii QPllsV 7
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ It is very hard to be sure. There may be several impediments to the evolution of a technical civilization. Planets may be rarer than we think. Perhaps the origin of life is not so easy as our laboratory experiments suggest. Perhaps the evolution of advanced life forms is improbable. Or it may be that complex life forms evolve more readily, but intelligence and technical societies require an unlikely set of coincidences - just as the evolution of the human species depended on the demise of the dinosaurs and the ice age recession of the forests in whose trees our ancestors screeched and dimly wondered. Or perhaps civilizations arise repeatedly, inexorably, on innumerable planets in the Milky Way, but are generally unstable; so all but a tiny fraction are unable to survive their technology and succumb to greed and ignorance, pollution and nuclear war. It is possible to explore this great issue further and make a crude estimate of N, the number of advanced civilizations in the galaxy. We define an advanced civilization as one capable of radio astronomy. Th is is, of course, a parochial if essential definition. There may be countless worlds on wh ich the inhabitants are accomplished linguists or superb poets but indifferent radio astronomers. We will not hear from them. N can be written as the product or multiplication of a number of factors, each a kind of filter, every one of which must be sizable for there to be a large number of civilizations: • Ns, the number of stars in the Milky Way galaxy. • fp, the fraction of stars that have planetary systems. • ne, the number of planets in a given system that are ecologically suitable for life. • fl, the fraction of otherwise suitable planets on which life actually arises. • fi, the fraction of inhabited planets on which an intelligent form of life evolves. • fc, the fraction of planets inhabited by intelligent beings on which a communicative techn ical civilization develops . • fl, the fraction of planetary lifetime graced by a technical civilization. Written out, the equation reads N = Ns • jjJ • ne • fl ·ft · Jc· fL (7) All of the f's are fractions, having values between O and 1; they will pare down the large value of Ns. To derive N we must estimate each of these quantities. We know a fa ir amount about the early factors in the equation, the number of stars and planetary systems. We know very little about the later factors, concerning the evolution of intelligence or the lifetime of technical societies. In these cases our estimates will be little better than guesses. I invite you, if you disagree with my estimates below, make your own choices and see what implications your alternative suggestions have for the number of advanced civilizations in the galaxy. One of the great virtues of this equation, due to Frank Drake of Cornell, is that it involves subjects ranging from stellar and planetary astronomy to organic chemistry, evolutionary biology, history, politics and abnormal psychology. Much of the Cosmos is in the span of the Drake equation. UNCLASSIFIED//FQA QFFICiIPL. !Iii 011! Y 8
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ We know Ns, the number of stars in the Milky Way galaxy, fairly well, by careful counts of stars in a small but representative region of the sky. It is a few hundred billion; some recent estimates place it at 4 x 10 11 . Very few of these stars are of the massive short lived variety that squander their reserves of thermonuclear fuel. The great majority have lifetimes of billions or more years in which they are shining stably, providing a suitable energy source for the energy and evolution of life on nearby planets. There is evidence that planets are a frequent accompaniment of star formation: in the satellite systems of Jupiter, Saturn and Uranus, which are like miniature solar systems; in theories of the origin of the planets; in studies of double stars; in observations of accretion disks around stars; and is some preliminary investigations of gravitational perturbations of nearby stars. 1 Many, perhaps even most, stars may have planets. We take the fraction of stars that have planets, fp, as roughly equa l to 1/3. Then the total number of planetary systems in the galaxy would be Ns fp ~ 1.3 x 10 11 (the symbol ~ means "approximately equal to"). If each system were to have about ten planets, as ours does, the tota l number of worlds in the galaxy would be more than a trillion, a vast arena for the cosmic drama. In our own solar system there are several bodies t hat may be suitab le for life of some sort: the Earth certainly, and perhaps Mars, Titan and Jupiter. Once life originates, it tends to be very adaptable and tenacious. There must be many different environments suitable for life in a given planetary system. But conservatively we choose ne=2. Then the number of planets in the galaxy su itable for life becomes Ns fp ne ~3 x 10 11 . Experiments show that under the most common cosmic conditions the molecular basis of life is read ily made, the building blocks of molecules able to make copies of themselves. We are now on less certain grounds; there may, for example, be impediments in the evolution of the genetic code, although I think this is unlikely over billions of years of primeval chemistry . We choose fl~ 1/3, implying a total number of planets in the Milky Way on which life has arisen at least once as Ns fp ne fl~ 1 x 10 11 , a hundred billion inhabited worlds . That in itself is a remarkable conclusion. But we are not yet fin ished. The choices of fi and fc are more difficult. On the one hand, many individually unlikely steps had to occur in biologica l evolution and human history for our present intelligence and technology to develop. On the other hand, there must be quite different pathways to an advanced civilization of specified capab ilities. Considering the apparent difficulty in the evolution of large organisms, represented by the Cambrian explosion, let us choose fix fc = 1/100, meaning that only 1 per cent of planets on wh ich life arises actually produce a technical civilization. This estimate represents some middle ground among the varying scientific options. Some think that the equivalent of the step from the emergence of trilobites to the domestication of fire goes like a shot in all planetary systems; others th ink t hat, even given ten or fifteen billion years, the evolution of a technical civi lization is unlikely. This is not a subject on wh ich we can do much experimentation as long as our investigations are limited to a single planet. Multiplying 1 Carl Sagan was writ ings these lines back in the 1970's, when no extrasolar planets had been discovered yet. The first such discovery occurred in 1995, wh en Michel Mayor and Didier Queloz, working at the "Observatoire de Haute Provence" in France, discovered the first extrasolar planet orbiting the nearby star 51 Peg. This first extrasolar planet was hence named 51 Peg B. Many more extrasolar planets were discovered around nearby stars ever since . As of Apri l 2009, 347 extrasolar planets (exoplanets) are listed in the Extrasolar Planets Encyclopaed ia. UNCLASSIFIED/ /POil Offl@IAL W&i 9Nls¥ 9
UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ these factors together, we find Ns fp ne fl fi fc ~ 1 x 10 9 , a billion planets on which technical civilizations have arisen at least once. But that is very different from saying that there are a billion planets on which technical civilizations now exist. For this we must also estimate fl. What percentage of the lifetime of a planet is marked by a technical civil ization? The Earth has harbored a technical civilization characterized by radio astronomy for only a few decades out of a lifetime of a few billion years. So far, then, for our planet fl is less than 1/10 8 , a millionth of a percent. And it is hardly out of the question that we might destroy ourselves tomorrow. Suppose this were a typical case, and the destruction so complete that no other technical civilization - of the human or any other species - were able to emerge in the five or so billion years remaining before the Sun dies. Then Ns fp ne fl fi fc fl ~ 10, and, at a given time there would be only a tiny smattering, a handful, a pitiful few technical civilizations in the galaxy, the steady state number maintained as emerging societies replace those recently self-immolated. The number N might be even as small as 1 if civilizations tend to destroy themselves soon after reaching a technological phase; there might be no one for us to talk with but ourselves. And that we do but poorly. Civilizations would take billions of years of tortuous evolution, and then snuff themselves out in an instant of unforgivable neglect. But consider the alternative, the prospect that at least some civilizations learn to live with technology; that the contradictions posed by the vagaries of past brain evolution are consciously resolved and do not lead to self destruction; or that, even if major disturbances occur, they are reveres in the subsequent billions of years of biological evolution. Such societies might live to a prosperous old age, their lifetimes measured perhaps on geological or stellar evolutionary time scales. If 1 percent of civilizations can survive technological adolescence, take the proper fork at this critical historical branch point and achieve maturity, then fl ~ 1/100, N ~ 10 7 , and the number of extant civilizations in t he galaxy is in the millions. Thus, for all our concern about t he possible unrel iabi lity of our estimates of the early factors in the Drake equation, which involve astronomy, organic chemistry and evolutionary biology, the principal uncertainty comes to economics and politics and what, on Earth, we call human nature. It seems fairly clear that if self-destruction is not the overwhelmingly preponderant fate of galactic civilizations, then the sky is softly humming with messages from the stars. These estimates are stirring . They suggest that the receipt of a message from space is, even before we decode it, a profoundly hopeful sign. It means that someone has learned to live with high technology; that it is possible to survive technological adolescence. This alone, quite apart from the contents of the message, provides a powerful justification for the search for other civilizations. 4. The Drake Equation is Over-Simplified In the nearly fifty years (1961-2009) elapsed since Frank Drake proposed his equation, a number of scientists and writers tried to find out which numerical values of its seven independent variables are more real istic in agreement with our present-day knowledge. Thus there is a considerable amount of literature about the Drake equation nowadays, and, as one can easily imagine, the results obtained by the various authors largely differ from one another. In other words, the value of N, that various authors obtained by different assumptions about the astronomy, the biology and the sociology implied by the Drake equation, may range from a few tens (in the pessimist's view) to some UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPII.¥ 10
UNCLASSIFIED/ /FOR OFFIEil.t.k W~i Q~lls¥ million or even billions in the optimist's opinion. A lot of uncertainty is thus affecting our knowledge of N as of 2010. In all cases, however, the final result about N has always been a sheer number, i.e., a positive integer number ranging from 1 to millions or billions. This is precisely the aspect of the Drake equation that th is author regarded as " too simpl istic" and improved mathematically in his paper #IAC-08-A4.1.4, entitled "The Statistical Drake Equation" and presented on October 1st , 2008, at the 59 th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29th thru October 3rd, 2008. That paper is attached herewith as Appendix B. Newcomers to SETI and to the Drake equation, however, may find that paper too difficult to be understood mathematically at a first reading. Thus, I shall now expla in the content of that paper "by speaking easily." I thank the reader for his or her attention . 5. The Statistical Drake Equation We start by an examp le. Consider the first independent variable in the Drake equation (7), i. e., Ns, the number of stars in the Milky Way galaxy. Astronomers tell us that approximately there should be about 350 millions stars in the galaxy. Of course, nobody has counted (or even seen in the photographic plates) all the stars in the galaxy! There are too many practical difficulties preventing us from doing so: just to name one, the dust clouds that don't allow us to see even the Galactic Bulge (i.e. the central region of the galaxy) in the visible light (although we may "see it" at radio frequencies like the famous neutral hydrogen line at 1420 MHz). So, it doesn't make any sense to say that Ns = 350 x 10 6, or, say (even worse) that the number of stars in the galaxy is (say) 354,233,321, or similar fanciful exact integer numbers. That is just silly and non-scientific. Much more scientific, on the contrary, is to say that the number of stars in the galaxy is 350 million plus or minus, say, 50 millions (or whatever values the astronomers may regard as more appropriate, since this is just an example to let the reader understand the difficulty). Thus, it makes sense to REPLACE each of the seven independent variables in the Drake equation (7) by a MEAN VALUE (350 millions, in the above example) PLUS OR MINUS A CERTAIN STANDARD DEVIATION (SO millions, in the above example) . By doing so, we have made a great step ahead: we have abandoned the too-simplistic equation (7) and replaced it by something more sophisticated and scientifically more serious: the STATISTICAL Drake equation. In other words, we have transformed the classical and simplistic Drake equation (7) into an advanced statistica l tool for the investigation of a host of facts hardly known to us in detail. In other words still: • We replace each independent variable in (7) by a RANDOM VARIABLE, labeled D, (from Drake). • We assume that the MEAN VALUE of each D, is the same numerical value previously attributed to the corresponding independent variable in (7). • But now we also ADD A STANDARD DEVIATION cr 0 ; on each side of the mean value, that is provided by the knowledge gathered by scientists in each discipline encompassed by each D,. UNCLASSIFIED//FQA QFFICilOL. !!ii ONI X 11
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Having so done, the next question is: How can we find out the PROBABILITY DISTRIBUTION for each D; ? For instance, shall that be a Gaussian, or what? This is a difficult question, for nobody knows, for instance, the probability distribution of the number of stars in the galaxy, not to mention the probability distribution of the other six variables in the Drake equation (7). There is a brilliant way to get around this difficulty, though. We start by excluding the Gaussian because each variable in the Drake equation is a POSITIVE (or, more precisely, a non-negative) random variable, while the Gaussian applies to REAL random variables only. So, the Gaussian is out. Then, one might consider the large class of well-studied and positive probability densities called "the gamma distributions," but it is then unclear why one should adopt the gamma distributions and not any other. The solution to th is apparent conundrum comes from Shannon's Information Theory and a theorem that he proved in 1948: "The probability distribution having maximum entropy ( = uncertainty) over any FINITE range of real values is the UNIFORM distribution over that range," This is proven in Appendix A of the present document. So, at this point, we assume that each of the seven D; in (7) is a UNIFORM random variable, whose mean value and standard deviation is known by the scientists working in the respective field (let it be astronomy, or biology, or sociology). Notice that, for such a uniform distribution, the knowledge of the mean value Po; and of the standard deviation u 0 , automatically determines the RANGE of that random variable in between its lower (called a;) and upper (called b;) limits: in fact these limits are given by the equations (8) (the "surprising" factor ✓ 3 in the above equations comes from the definitions of mean value and standard deviation: please see equations (12), (15) and (17) in Appendix B for the relevant proof). So the uniform distribution of each random variable D; is perfectly determined by its mean value and standard deviation, and so are all its other properties. The next problem is the following: OK, since we now know everything about each uniformly distributed D;, what is the probability distribution of N, given that N is the product (7) of all the D 1 ? In other words, not only do we want to find the analytical expression of the probability density function of N, but we also want to relate its mean value f-lN to all mean values μ of the D;, and its standard deviation to all standard deviations of the D;. , u N u , 0 0 12 UNCLASSIFIED/ fFOR: QFFICiIO L. lalii ODIL.¥
UNCLASSIFIED/ /FOR OFFI&I.t.k W~i Q~lk¥ This is a difficult problem. It occupied the author's mind for no less than about ten years (1997 -2007). It is actually an ANALYTICALLY UNSOLVABLE problem, in that, to the best of this author's knowledge, it is IMPOSSIBLE to find an analytic expression for any FINITE PRODUCT of uniform random variablesD; . This result is proven in Sections 2 thru 3.3 of Appendix B (unfortunately!) . 6. Solving the Statistical Drake Equation By Virtue of the Central Limit Theorem (CLT) of Statistics The solution to the problem of finding the analytical expression for the probability density function of N in the statistica l Drake equation was found by this author in September 2007. The key steps are the following: • Take the natural logs of both sides of the statistical Drake equation (7). This changes the product into a sum. • The mean values and standard deviations of the logs of the random variables D; may all be expressed analytically in terms of the mean values and standard deviations of the D; . • Recall the Central Limit Theorem (CLT) of statistics, stating that (loosely speaking) if you have a SUM of independent random variables, each of which is ARBITRARILY DISTRIBUTED (hence, also including uniformly distributed), then, when the number of terms in the sum increases indefinitely (i.e. for a sum of random variables infinitely long) .. . the SUM RANDOM VARIABLE TENDS TO A GAUSSIAN. • Thus, the natural log of N tends to a Gaussian. • Thus, N tends to the LOGNORMAL DISTRIBUTION. • The mean value and standard deviations of this lognormal distribution of N may all be expressed analytically in terms of the mean values and standard deviations of the logs of the D, already found previously. This result is fundamental. All the relevant equations are summarized in the following Table 1. This table is actually the same as Table 2 of the author's original paper IAC-08-A4.1.4, entitled "The Statistical Drake Equation" and presented by him at the International Astronautical Congress (IAC) held in Glasgow, UK, on October l5t, 2008. This orig inal paper is reproduced in Appendix B. To sum up, not only is it found that N approaches the completely known lognormal distribution for an INFINITY of factors in the statistical Drake equation (7), but the way is paved to further applications by removing the cond ition that the number of terms in the product (7) must be FINITE. UNCLASSIFIED/ /FOR 8FFI&l.t.k Wii QNk¥ 13
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ This possibility of ADDING ANY NUMBER OF FACTORS IN THE DRAKE EQUATION (7) was not envisaged, of course, by Frank Drake back in 1961, when "summarizing" the evolution of life in the galaxy in SEVEN simple STEPS. But today, the number of factors in the Drake equation should already be increased: for instance, there is no mention in the original Drake equation of the possibility that asteroidal impacts might destroy the life on Earth at any time, and this is because the demise of the dinosaurs at the K/T impact had not been yet understood by scientists in 1961, and was so only in 1980! In practice, the number of factors should INCREASE as much as necessary in order to get better and better estimates of N as long as our scientific knowledge increases. This is called the "Data Enrichment Principle" and believe should be the next important goal in the study of the statistical Drake equation. Finally, a numerical example explaining how the statistical Drake equation works in the practice will be given in the next section. UNCLASSIFIED/ /rOR orr1e1At HSE OrtLY 14
UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ Table 1. Summary of the Properties of the Lognormal Distribution That Applies to the Random Variable N = Number of ET Communicating Civilizations in the Galaxy Random variable N = number of communicating ET civilizations in galaxy Probab ility distribution Loq norma l Probabi lity density function (1n(11h,J' 1 1 -~ (n ;::-: 0) n 27!<:T JN (n)= - · & e a' Mean va lue - (N )= eμe 2 Variance a 1 = e 2 μ ea' (ea' - I) a' a N = eμ e 2 .Jea 2 - 1 Standard deviation All the moments, i. e. k-th moment 2 a' k · - 2 (N k)= ekμ e _ _ ~I - a- 2 Mode ( = abscissa of the log normal peak) n nnde = npeak - e e a' f ( ) Value of the Mode Peak 1 -p 2 N n lTl)dc = & ·e ·e 2,r (l Median (= fifty-fifty probability value for rred ian = m = e" N) Skewness K ( , ) e-6pe- 3a' _ 3 _ = ea + 2 (K4)¾ ., lne3a2 + 3e2a' +6ea' +61 k' - K ,, ., ., "} Kurtosis _ 4_ = e4a- + 2 e3a- + 3 e-u- - 6 (K2)2 Expression of 1-1 in terms of the lower (a;) μ = ± (Y;) = ± b;[in(b;) - 1]- a;[in (a;) -1] and upper (b ;) limits of the Drake i =I i= I b; - a; uniform input random va riables 0 ; Expression of <:T 2 in terms of the lower (a;) 7 7 a;b; [ln (b; )- in(a;)f a 2 =La~ = L l - and upper (b;) limits of the Drake (b; -a} i= I i =I uniform input random variables 0 ; 7. An Example Explaining the Statistical Drake Equation To understand how things work in practice for the statistical Drake equation, please consider the following table 2. It is made up of three columns: • The first column on the left lists the seven input sheer numbers that also become • The mean values (middle column). • Finally the last column on the right lists the seven input standard deviations. UNCLASSIFIED/ 1re1t OFFl@IAL WSE 9Nk¥ 15
UNCLASSIFIED/ /FOR OFFIEil.t.k W&i Q~lls¥ The bottom line is the classical Drake equation (7). We see that, for this particular set of seven inputs, the classical Drake equation (i.e. the product of the seven numbers) yields a total of 3500 communicating extraterrestrial civilizations existing in the galaxy right now. rs := 350 -10 9 s := s 0 10 μfp := fp fp := 100 ofp := 100 l ne := 1 μne := ne crne := - /3 0 fl := - μfl := fl crfl := ~ 100 100 fi :=~ μfi := fi on := ~ 100 100 0 fc:= - μfc := fc crfc := ~ 100 100 fL := 10000 crtL := 1000 μfl. := tL 1010 1010 _ 1 := Ks -fp -ne-fl -fi .fc .ff, = 3500 Table 2. Input Values (i.e. mean values and standard deviations) for the Seven Drake Uniform Random Variables Di . The first column on the left lists the seven input sheer numbers that also become the mean values (middle column). Finally the last column on the right lists the seven input standard deviations . The bottom line is the classical Drake equation (7). The statistical Drake equation, however, provides a much more articulated answer than just the above sheer number N = 3500. In fact, a MathCad code written by this author and capable of performing all t he numerical calculations required by the statistical Drake equation for a given set of seven input mean va lues plus seven input standard deviations, yields for N the lognormal distribution (thin curve) plotted in Figure 2. We see immediately that the peak of this thin curve (i.e. the mode) falls at about n rmde;;;; npeak = eμ e-a' ""250 (this is equation (99) of Appendix B), while the median (fifty fifty value spl itting the lognormal density in two parts with equal undergoing areas) falls at about nm,d ian;;;; eμ ""1740 . These seem to be smaller values than N = 3500 provided by the classical Drake equations, but it's a wrong impression due to a poor "intuitive" understanding of what statistics is! In fact, neither the mode nor the median are the " really important" values: the really important value for N is the MEAN VALUE! Now if you look at t he thin curve in Figure 2 below (i.e. the lognorma l distribution arisin g from the Central Limit Theorem), you see that this curve has a LONG TAIL ON THE RIGHT! In other words, it does NOT immediately go down to nearly zero beyond the peak of the mode. Thus, when you actually compute the mean value, you should not be too UNCLASSIFIED// POR OPPICll<L ti.!! er~LV 16
UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ (]'' surprised to find out that it equals (N) = e 11 e 2 ::::: 4589 .559 ~ 4590 communicating civilizations now in the galaxy. This is the important number, and it is HIGHER than the 3500 provided by the classical Drake equation. Thus, in conclusion, THE STATISTICAL EXTENSION of the classical Drake equation INCREASES OUR HOPES to find an extraterrestria I civi Iization ! PROBABILITY DENSITY FUNCTION OF N 1000 2000 3000 4000 N = Number of ET Civilizations in Galaxy Figure 2. Comparing the Two Probability Density Functions of the Random Variable N Found (1) Without Resorting to the CLT at All (thick curve) and (2) Using the CLT and the Relevant Lognormal Approximation (thin curve). Even more so our hopes are increased when we go on to consider the standard deviation associated with the mean value 4590. In fact, the standard deviation is given ,,., by equation (97) of Appendix B. This yields er"' = e 11 e 2 ✓ e" 2 -1 = 11195 and so the expected number of N may actually be even much higher than the 4590 provided by the mean value alone! The "upper limit of the one-sigma confidence interval" (as statisticians call it), i.e. the sum 4590+11195 = 15,785, yields a higher number still! (Note: the "lower limit of the one-sigma confidence interval is ZERO because the lognormal distribution is POSITIVE (or, more correctly, non-negative)). Finally, the reader should note that the thick curve depicted in Figure 2 is just the NUMERICAL solution of the statistical Drake equation for a FINITE number of 7 input factors. Figure 2 actually shows that this curve "is well interpolated" by the lognormal distribution (thin curve), i.e., by the neat analytical expression provided by the Central Limit Theorem for an INFINITE number of factors in the Drake equation. That is, in conclusion, Figure 2 visually shows that taking 7 factors or an infinity of factors "is almost the same thing" already for a value as small as 7. UNCLASSIFIED/ /FOil OFFI@IAb W&i QNI.¥ 17
UNCLASSIFIED/ /FOR OFFIEil.t.k Wii Q~lls¥ 8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical Drake Equation Having solved the statistical Drake equation by finding the lognormal distribution, we are now in a position to solve the ET-DISTANCE problem by resorting to statistics again, rather than just to the purely deterministic Distance Law (5), as we did in Section 2. This is "scientifically more serious" than just the purely deterministic Distance Law (5) inasmuch as the new statistical Distance Law will yield a PROBABILITY DENSITY for the Distance, with the relevant mean value and standard deviation . In other words, the Distance Law (5) itself becomes a random variable whose probability distribution, mean value and standard deviation must be computed by "replacing" into (5) the fact that N is now known to follow the lognormal distribution . This is mathematically described in detail in Section 7 of Appendix A. The important new result is the PROBABILITY DENSITY FOR THE DISTANCE, the equation of which is (9) holding for r ;,: O. This is equation (114) of Appendix B. Starting from this equation, the MEAN VALUE OF THE random variable ET_DISTANCE is computed as J.I cr2 (Er_Distance) =Ce 3 e ""jg (10) which is equation (119) of Appendix B, and finally the ET_DISTANCE STANDARD DEVIATION _!!. a2Kr2 - 3 18 9 CJ ET_Distanu: - C e e e - l (11) which is equation ( 123) of Appendix B. Of course, all other descriptive statistical quantities, such as moments, cumulants etc. can be computed upon starting from the probability density (9), and the resu lt is Table two hereafter, that is Table 3 of Appendix B. Finally, to complete this section, as well as this "introduction to the statistical Drake equation," the numerical values that equations (10) and (11) yield for the Input Table 1 are determined. They are, respectively: _}!_ CTl r,11ea11 _ m l ue = Ce 3 e 18 ;::: 2,670 light years (12) UNCLASSIFIED/ /FOR OFFIEil.t.k Wii QNls¥ 18
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ which is equation (153) of Appendix B, and I' (12 ~ a ET_Di stono, =C e 3 e •8 ~ e 9 - 1 a:: 1,309 light years (13) which is equation (154) of Appendix B. UNCLASSIFIED/ /FOR &FFIGIAk lalii ODIL:¥ 19
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Table 2. Summary of the Properties of the Probability Distribution That Applies to the Random Variable ET_Distance Yielding the (average} Distance Between Any Two Neighboring Communicating Civilizations in the Galaxy Random variable ET_Distance between any two neighboring ET civilizations in galaxy assuming they are UNIFORMLY distributed throughout the whole qalaxy volume. Probability distribution Unnamed Probability density function _(•{ 6 R [ata '.I l,Gola')]-μ J 1 3 I 2u 2 fET_Distana,(r) =-;. · Ji; CT ·e Numerical constant C related to the Milky Way size C =3 6 R 8alruy h Gala.,y "" 28,845 light years Mean value μ u2 (Er_Distance) = C e-3 e 18 Variance 2 2 2 - 2 2 a -3p 9 [ a 9 - l CTET_Distance - C e e e I Standard deviation 2 _ji_ er ✓ a - 3 18 9 _ CTET_Distance - C e e e l All the moments, i.e. k-th moment k2,u2 - k!!. (Er_Dis tan eek) = c k e 3 e 18 Mode ( = abscissa of the log normal peak) _ji_ er 9 rnnde = rpeak = Ce 3 e Value of the Mode Peak Peak Value of fET_Distanu:(r) = a' !!. 3 - · e 3 . e 18 = fET_Distana,Cr,rnde) = cfi; CT Median ( = fifty-fifty probability value for N) _ji_ ~dian = m = Ce 3 Skewness ,,., 5,,.2 ,,., ] e-μ [ e 2 - 3e 18 +2e 6 __!!i_ = 3 3 (K 4 )2 8a 2 5a 2 4a 2 2 a 2a 2 ]2 C3 [ e9 -4e- 9 -3e9 +12 e 3 - 6e- 9 Ku rtosis 4 a 2 a' 2 a 2 - - K4 _ 9 +2e 3 +3e 9 -6 (K2)2 - e Expression of μin terms of the lower (ai) and upper (bi) limits of the Drake uniform input random variables Di μ= I (r;) = I b; [ln (b;)- 1]- a;[ln(a;)- 1] i=I ;~1 b; - a; Expression of CT 2 in terms of the lower (ai) and upper (bi) limits of t he Drake uniform input random variables Di 7 7 a;b; [ln (b; )- ln(a;)f CT 2 = Io} = Ii (b; - a; ) 2 i= I i=I UNCLASSIFIED/ J 'FQA QFFICiIPL. !Iii 011! Y 20UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ UNCLASSIFIED// POil OFFI@IAb W&i 91'11.¥ 21
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ It is clarifying to draw the graph of the ET_Distance probability density (9): DISTANCE OF NEAREST Ef_ CIVILIZA TION 5.63 ·10- 20 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 ET _Distance from Earth (light years) Figure 3. The Probability of Finding the Nearest Extraterrestrial Civilization at the distance r From Earth (in light years) if the Values Assumed in the Drake Equation are Those Shown in Input Table 1. The relevant probability density function fET_Disian.,( r) is given by equation (9). Its mode (peak abscissa) equals 1933 light years, but its mean value is higher since the curve has a long tail on the right : the mean value equals in fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR SETI, inasmuch as the nearest ET galaxy civilization might lie at just 1 sigma = 2670-1309 = 1361 light years from us. From Figure 3 we see that the probability of finding extraterrestrials is practically zero up to a distance of about 500 light years from Earth. Then it starts increasing with the increasing distance from Earth, and reaches its maximum at _l!_ ~ r=de = r peak = C e 3 e 9 :::: 1,933 light years. (14) This is the MOST LIKELY VALUE of the distance at which we can expect to find the nearest extraterrestrial civilization. It is not the mean va lue of the probability distribution (9) for fET_Di stan.,( r). In fact, the probability density (9) has an infinite tail on the right, as clearly shown in Figure 3, and hence its mean value must be higher than its peak value. As given by (10) and (12), its ....!:!.. 0'2 mean value is r,,,en,,_mlue = Ce 3 e 18 c::,2670 light years. This is the MEAN (va lue of the) DISTANCE at which we can expect to find extraterrestria ls. UNCLASSIFIED/ ; 'FQA QFFICiIPL. !Iii 011! Y 22
UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ After having found the above two distances (1933 and 2670 light years, respectively), the next natural question that arises is: "what is the range, back and forth around the mean value of the distance, within which we can expect to find extraterrestria ls with "the highest hopes?" The answer to this question is given by the notion of standard deviation that we already found to be given by (11) and (13), I' 0'2 ~ CT ET_Distance = Ce 3 e •8 V e 9 -1 ""1309 light years . More precisely, this is the so-ca lled 1-sigma (distance) level. Probability theory then shows that the nearest extraterrestrial civilization is expected to be located within this ra nge, i.e. within the two distances of (2670-1309) = 1361 light years and (2670+1309) = 3979 light years, with probability given by the integral of fET_oistance(r) taken in between these two lower and upper limits, that is: i 3979 1igh1 years fET Di siance (r) dr:::: 0.75 = 75 % (15) l36 1ligbtycars - In plain words: with 75 percent probability, the nearest extraterrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by table 1. If we change those input values, then all the numbers change again, of course. 9. The "Data Enrichment Principle" as the Best CLT Consequence Upon the Statistical Drake Equation (Any Number of Factors Allowed) As a fitting climax to all the statistical equations developed so far, let us now state our "DATA ENRICHMENT PRINCIPLE." It simply states that "The Higher the Number of Factors in the Statistical Drake equation, The Better." Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the normal distribution when the number of terms in the sum (4) approaches infinity. And this is the case, indeed. 10. Conclusions We have sought to extend the classical Drake equation to let it encompass Statistics and Probability. This approach appears to pave the way to future, more profound investigations intended not only to associate "error bars" to each factor in the Drake equation, but especially to increase the number of factors themselves. In fact, this seems to be the only way to incorporate into the Drake equation more and more new scientific information as soon as it becomes available. In the long run, the Statistical Drake equation might just become a huge computer code, growing in size and especially in the depth of the scientific information it contains. It would thus be Humanity's first "Encyclopaedia Galactica." UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPlls¥ 23
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Unfortunately, to extend the Drake equation to Statistics, it was necessary to use a mathematical apparatus that is more sophi sticated than just the simple product of seven numbers. UNCLASSIFIED/ /FOR 8FFIGl.t.k W&i QPII.¥ 24
1 UNCLASSIFIED/ /FOR OFFIEil.t.k W~i Q~lls¥ Appendix A: Proof of Shannon's 1948 Theorem Stating That the Uniform Distribution is the "Most Uncertain" One Over a Finite Range of Values Information Theory was initiated by Claude Shannon (1916-2001) in his well-known 1948 two papers: Rqrutted mm commi:ms from B • 1-S,'l• st Ii •ral Jour:!li, \ GI. 1 . pp. 9--!13 , 62J..,iS , July, Occober. I A Mathematical Theory of Communication By C. E SHANNO In this Appendix, we wish to draw attention to a couple of theorems that Shannon proves on pages 36 and 37 of his work, and read, respectively (note that Shannon omits the upper and lower limits of all integrals in the first theorem: they are minus infinity and plus infinity, respectively): 5. Letp(x) bea one-dimensionaldis :i ti The onno p(x) ginng ammcimumentropy 1:.ubjec-tto the c dition tha e dan:l devia on ofx be fixed at a i- G ian. To show this we must maximize H(x) = - f p x) logp(x) d <T- = j p x- dx and 1 =/ p x)dx as cons •. This requi~ . by calculu~ of variations maxunizing / [- p x)logp(x) >.p(x 1p(x)] dx. e condition or lus is - 1- logp(x •ng the constllllf5 to sat' the and UNCLASSIFIED/ /FOR: QFFIGI0ls Uii 0111 X 25
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ 7. If x is limited to a half line (p(x) = 0 for x < 0) and the fin.t moment of x is ihted at a: a = f p (x dx then the maximwn en opy OCCW1! when p(x) =,! - f.r/al a and 1s equal to logea. Now, we wish to point out that there is a third possible case, other than the two given by Shannon. This is the case when the probabi lity density function p(x) is limited to a FINITE INTERVAL a::; x::; b. This is obviously the case with any physical POSITIVE ra ndom variable, such as a distance, or the number N of extraterrestrial communicating civilizations in the,". And it is easy to prove that for any such finite random variable the maximum entropy distribution is the UNIFORM distribution over a -5, x -5, b. Shannon did not bother to prove this simple theorem in his 1948 papers since he probably regarded it as too trivial. But we prefer to point out this theorem since, in the language of the statistica l Drake equation, it sounds like: "Since we don't know what the probability distribution of any one of the Drake random variables D; is, it is safer to assume that each of them has the maximum possible entropy over a; -5,x-5, h; , i. e., that D; is UNIFORM LY distributed there. The proof of th is theorem is along the same lines as for the previous two cases discussed by Shannon: We start by assuming t hat a; -5, x -5, b; . We then form the linear combination of the entropy integral plus the normalization condition for D; where i is a Lagrange multipli er. Performing the variation, one finds - Iogp(x)- 1+1 = 0 that is: p(x)= e ,1,- i . App lying the normalization condition (constraint) to the last expression for p(x) yields b, ( ) f. b; ,l,-1 2-1 f.b; ,1,-1 ( ) I = f. p x dx = e dx = e dx = e b; - a,. a1 a1 a1 that yields ,l,-1 1 e =- - h; - a; UNCLASSIFIED/ /POlt Offl@IAL WSi 9Nk¥ 26
UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ and finally p(x) = - 1 - with a; s x s b; b; - a; showi ng that the maximum-entropy probability distribution over any FINITE int erval a; sxsb; is the UNIFORM distrib ution . UNCLASSIFIED/ /FOR OFFI&IAk Wlii QNk¥ 27
UNCLASSIFIED/ /FOR OFFIEil.t.k W&i Q~lls¥ Appendix B: Original Text of the Author's Paper #IAC-08 - A4.1.4 Titled the Statistical Drake Equation IA C-08-A4.1.4 THE STATISTICAL DRAKE EQUATION Claudio Maccone Co-Vice Chair, SETI Permanent Study Group, International Academy ofAstronautics Address: Via Martorelli, 43 - Torino (Turin) 10155 - Italy URL: http://www.maccone.com / - E-mail: clmaccon@libero.it ABSTRACT. We provide th statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake eq uation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation," The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. u01mal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of th CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: l) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary Nin the Drake equation. The standard deviation , mode, and all the moments of this lognormal N are found also. 2) The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution fo r each factor. This is both physically realistic and practicalJ y very useful, of course. 3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function , apparently previously unknown and dubbed "Maccone distribution" by Paul Davies. 4) DATA ENRICHMENT PRINCIPLE. It should be noticed that ANY positive number of random vai'iables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation we call the "Data Enrichment Principle", and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. Finally, a practical example is given of how our statistical Drake equation works numerically. We work out in detail the case where each of the seven random variables is uniformly distributed around its own mean value and has a given standard deviation. For instance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 350 billions with a standard deviation of (say) I billion. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows UNCLASSIFIED/; P"OR OP"P"lelltt li!I! er•tv 28
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ that the mean value of the lognormal random variable N is actually of the same order as the classical N given by the ordinary Drake equation, as one might expect from a good statistical generalization. 1. INTRODUCTION The Drake equation is a now famous result (see ref. [1] for the Wikipedia summary) in the fields of SETI (the Search for ExtraTetTestial Intelligence, see ref. [2]) and Astrobiology (see ref. [3]). Devised in l 960, the Drake equation was the first scientific attempt to estimate the number N of ExtraTerrestrial civilizations in the Galaxy with which we might come in contact. Frank D. Drake (see ref. [4]) proposed it as the product of seven factors: N=Ns-fp-ne·fl·fi·fc·fL. (1) Where: I) Ns is the estimated number of stars in our Galaxy. 2) fp is the fraction (= percentage) of such stars that have planets. 3) ne is the number "Earth-type" such planets around the given star; in other words, ne is number of planets, in a given stellar system, on which the chemical conditions exist for life to begin its course: they are "ready for life," 4) fl is fraction(= percentage) of such "ready for life" planets on which life actually starts and grows up (but not yet to the "intelligence" level). 5) fl is the fraction (= percentage) of such "planets with life fom1S" that actually evolve until some form of "intelligent civilization" emerges (like the first, historic human civilizations on Earth). 6) Jc is the fraction (= percentage) of such "planets with civilizations" where the civilizations evolve to the point of being able to communicate across the interstellar distances with other (at least) similarly evolved civilizations. As far as we know in 2008, this means that they must be aware of the Maxwell equations governing radio waves, as well as of computers and radioastronomy (at least). 7) fl is the fraction of galactic civilizations alive at the time when we, poor humans, attempt to pick up their radio signals (that they throw out into space just as we have done since l 900, when Marconi started the transatlantic transmissions). In other words, fl is the number of civilizations now transmitting and receiving, and this implies an estimate of"how long will a technological civilization live?" that nobody can make at the moment. Also, are they going to destroy themselves in a nuclear war, and thus live only a few decades of technological civi lization? Or are they slowly becoming wiser, reject war, speak a single language (like English today), and merge into a single "nation", thus living in peace for ages? Or will robots take over one day making "flesh animals" disappear forever (the so-called "post-biological universe")? No one knows ... But let us go back to the Drake equation (1). In the fifty years of its existence, a number of suggestions have been put forward about the different numeric values of its seven factors. Of course, every different set of these seven input numbers yields a different value for N, and we can endlessly play that way. But we claim that these are like ... children plays! We claim the classical Drake equation (1), as we shall call it from now on to distinguish it from our statistical Drake equation to be introduced in the coming sections, well, the classical Drake equation is scientifically inadequate in one regard at least: it just handles sheer numbers and does not associate an error bar to each of its seven factors. At the very least, we want to associate an error bar to each D;. Well, we have thus reached STEP ONE in our improvement of the classical Drake equation: replace each sheer number by a probability distribution! The reader is now asked to look at the flow chart in the next page as a guide to this paper, please. 2. STEP 1: LETTING EACH FACTOR BECOME A RANDOM VARIABLE In this paper we adopt the notations of the great book "Probability, Random Variables and Stochastic Processes" by Athanasios Papoulis (1921-2002), now re-published as Papoulis-Pillai, UNCLASSIFIED/ ,'FOR: QFFIGI CL. Wii ODIL:¥ 29
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ ref. [5]. The advantage of this notation is that it makes a neat distinction between probabilistic (or statistical: it's the same thing here) variables, always denoted by capitals, from non-probabilistic (or "detenninistic") variables, always denoted by lower-case letters. Adopting the Papoulis notation also is a tribute to him by this author, who was a Fulbright Grantee in the United States with him at the Polytechnic [nstitute (now Polytechnic University) of New York in the years 1977-78-79. We thus introduce seven new (positive) random variables D; ("D" from "Drake") defined as D1 = Ns D2 =fp D 3 =ne D4 = fl (2) Ds =fl D6 =Jc D 7 =fL so that our STATISTICAL Drake equation may be simply rewritten as (3) Of course, N now becomes a (positive) random variable too, having its own (positive) mean value and standard deviation. Just as each of the D; has its own (positive) mean value and standard deviation ... ... the natural question then arises: how are the seven mean values on the right related to the mean value on the left? ... and how are the seven standard deviations on the right related to the standard deviation on the left? Just take the next step ... 3. STEP 2: INTRODUCING LOGS TO CHANGE THE PRODUCT INTO A SUM Products of random variables are not easy to handle in probability theory. It is actually much easier to handle sums of random variables, rather than products, because: 1) The probability density of the sum of two or more independent random variables is the convolution of the relevant probabifay densities (worry not about the equations, right now) . 2) The Fourier transform of the convolution simply is the product of the Fourier transforms (again, worry not about the equations, at this point) UNCLASSIFIED//EOR OFFICIO! !!SF ON! X 30UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ I 1. Introduction I I 2. Step 1: Letting each factor become a random I 2.1. Step 2: Introducing logs to change the product into a I I 2.2. Step 3: The transformation law of random variables. I I 3. Step 4: Assuming the easiest input distribution for each D1: the uniform distribution. I 3 .1. Step 5: A numerical example of the Statistical Drake equation with uniform distributions for the Drake random variables 0 ;. I 3.2. Step 6 : Computing the logs of the 7 uniformly distributed Drake random variables O; . I 3.3. Step 7: Finding the probability I density function of N, but only numerically not analytically. I DEAD END! I 4. The Central Limit Theorem (CLT) of Statistics. I s. LOGNORMAL distribution as the probability distribution of the number N of communicating ExtraTerrestrial Civilizations in the Galaxy. I 6. Compari ng the CLT r esults with the Non- CLT results, and discarding the Non-CLT approach. 7. DISTANCE to the nearest ExtraTerrestriai Civilization as a probability distribution ( Paul Davies dubbed that the Maccone distribution). I 7. 1 Classical, non-probabilistic derivation of the Distance to the nearest ET Civilization. 7.2 Probabilistic derivation of probability density function for nearest ET Civilization Distance. 7.3 Statistical properties of the distribution. 7.4 Numerical example of the distribution. I 8. DATA ENRICHMENT PRINCIPLE as the best CLT consequence upon the Drake equation : ;mx number of factors allowed for. UNCLASSIFIED/ ; 'FQA QFFICilPL. PP&i ODI! Y 31
UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ So, let us take the natural logs of both sides of the Statistical Drake equation (3) and change it into a sum: It is now convenient to introduce eight new (positive) random variables defined as follows: Y=ln(N) { (5) Y;=ln(D;) i=l,...,7. Upon inversion, the first equation of (5) yields the important equation, that will be used in the sequel (6) We are now ready to take STEP THREE. STEP 3: THE TRANSFORMATION LAW OF RANDOM VARIABLES So far we did not mention at all the problem: "which probability disttibution shall we attach to each of the seven (positive) random variables D; ?" It is not easy to answer this question because we do not have the least scientific clue to what probability distributions fit at best to each of the seven points listed in Section 1. Yet, at least one trivial error must be avoided: claiming that each of those seven random variables must have a Gaussian (i.e. normal) distribution. In fact, the Gaussian distribution, having the well known bell-shaped probability density function (a ;,: o) (7) has its independent variable y ranging between --oo and oo and so it can apply to a real random variable Y only, and never to positive random variables like those in the statistical Drake equation (3). Period. Searching again for probability density functions that represent positive random variables, an obvious choice would be the gamma distributions (see, for instance, ref. [6]). However, we discarded this choice too because of a different reason: please keep in mind that, according to (5), once we selected a particular type of probability density function (pdt) for the last seven of equations (5), then we must compute the (new and different) pdf of the logs of such random variables. And the pdf of these logs certainly is not gamma-type any more. It is high time now to remind the reader of a certain theorem that is proved in probability courses, but, unfortunately, does not seem to have a specific name. It is the transformation law (so we shall call it, see, for instance, ref. [5]) allowing us to compute the pdf of a certain new random variable Y that is a known function Y = g(X) of another random variable X having a known pdf. In other words, if the pdf fx (x) of a certain random variable X is known, then the pdf fr(Y) of the new random variable Y, related to X by the functional relationship y = g(X) (8) can be calculated according to this rule: 1) First invert the corresponding non-probabilistic equation y = g(x) and denote by X; (y) the various real roots resulting from the this inversion. 2) Second, take notice whether these real roots may be either finitely- or infinitely-many, according to the nature of the function y = g(x). 3) Third, the probability density function of Y is then given by the (finite or infinite) sum (9) where the summation extends to all roots x;(Y) and lg' (x; (y)~ is the absolute value of the first derivative of g(x) where the i-th root x;(Y) has been replaced instead of x. Since we must use this transformation law to transfer from the D; to the Y; =ln(D;), it is clear that we need to start from a D; pdf that is as simple as possible. The gamma pdf is not responding to this need because the analytic expression of the transformed pdf is very complicated (or, at least, it looked so to this author in the first instance). Also, the gamma distribution has two free parameters in it, and this "complicates" its application to the various meanings of the Drake equation. Tn conclusion, we discarded the gamma distributions and confined UNCLASSIFIED/ /FOil OFFI@IAb W&i! QNI.¥ 32I UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ ourselves to the simpler uniform distribution instead, as shown in the nest section. 4. STEP 4: ASSUMING THE EASIEST PUT DISTRIBUTION FOR EACH D; : THE U IFORM DISTRIBUTION Let us now suppose that each of the seven D; is distributed UNIFORMLY in the interval ranging from the lower limit a; ~ 0 to the upper limit b; ~a;· This is the same as saying that the probability density function of each of the seven Drake random variables D; has the equation 1 funi rom, o. (x) = - - with O ::; a; s; x s; b; (10) - ' h;-a; as it follows at once from the normalization condition b f , funi rol'TlLD- (x )dx = 1 . (11) ti; I Let us now consider the mean value of such uniform D; defined by b; 1 f b' (unifom'LD;)= f xfunibniLo, (x)dx=--- xdx ai b; -a; a; 2 ]b; 2 2 = -- 1- .::._ = b; - a; = a; + b; [ b; - a; 2 a; 2 (b; - a;) 2 • By words (as it is intuitively obvious): the mean value of the uniform distribution simply is the mean of the lower plus upper limit of the variable range a -+b. ( unifonn_D ; ) = -'-- ' . (12) 2 In order to find the variance of the uniform distribution, we first need finding the second moment • 2) fb; 2 ( ) ( uniform_ Di = X fun ibrnLD, X dx 0 1 I ", , l x 3 ]b; h- 3 - a . 3 [ 3 = b-a . Lx-dx = b. -a . = 3(b. -;, .) I I I I I <l; I I _ (b; -a;)(a; +a;b; +b; ) _ a; +a;b; +b; - 3 (b; - a;) - 3 The second moment of the uniform distribution is thus D 2) a; +a;b; + b; ( un iti orm . = ---'---'-'-----'- (13) - I 3 From (12 and (13) we may now derive the variance of the uniform distribution c:r~n ifi>rnLD; = ( uniform_D ; 2 )-( uniform_ D ;) 2 a; ? + a;b; + b; 2 = ( 14) 3 4 12 Upon taking the square root of both sides of ( 14 ), we finally obtain the sta11dard deviation of the u11iform distribution : We now wish to perform a calculation that is mathematically trivial, but rather unexpected from the intuitive point of view, and very important for our applications to the statistical Drake equation. Just consider the two simultaneous equations ( 12) and (15) a . +b - ) uniform_ Di =~ l( (16) b- - a (T __ , __ , uniIDTITL D; - 2 ✓ 3 • Upon inverting this trivial linear system, one finds fa; = (uniform_D; )- ✓ 3 (Tuni Dn TLD; (17) lb; = ( uniform_D;) + ✓ 3 c:runifom 1..D, • This is of paramount importance for our application the Statistical Drake equation inasmuch as it shows that: if one (scientifically) assigns the mean value and standard deviation of a certai11 Drake random variable D;, then. the lower a11d upper limits of the relevant uniform distribution are given by the two equations (17), respectively. UNCLASSIFIED/ /POlt 9Pfl@IAL H§E 8Nk¥ 33
UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ In other words, there is a factor of ✓ 3 = 1.732 included in the two equations (17) that is not obvious at all to human intuition, and must indeed be taken into account. The application of this result to the Statistical Drake equation is discussed in the next section. 3.1 STEP 5: A NUMERICAL EXAMPLE OF THE STATISTICAL DRAKE EQUATION WITH UNIFORM DISTRIBUTIO S FOR THE DRAKE RANDOM VARIABLES D; The first variable Ns in the classical Drake equation (1) is the number of stars in our Galaxy. Nobody knows how many they are exactly (!). Only statistical estimates can be made by astronomers, and they oscillate (say) around a mean value of 350 billions (if this value is indeed correct!). This being the situation, we assume that our uniformly distributed random variable Ns has a mean value of 350 billions minus or plus a standard deviation of (say) one billion (we don't care whether this number is scientifically the best estimate as of August 2008: we just want to set up a numerical example of our Statistical Drake equation). ln other words, we now assume that one has: (uniform_D 1 ) = 350 ~10 9 { (18) a unilomLD, =l • 10 • Therefore, according to equations (17) the lower and upper limit of our uniform distribution for the random variable Ns=D1 are, respectively aNs = 1 uniform_D 1 ) - ✓ 3auni "'" rm O = 348.3-10 9 { \ . ,v - I 9 ( ( 9) bNs =(umform_D 1 ) + ,fj aunii:>rTTLD, =351.7 -10 Similarly we proceed for all the other six random variables in the Statistical Drake equation (3). For instance, we assume that the fraction of stars that have planets is 50%, i.e. 50/100, and thi s will be the mean value of the random variable fp=D 2 . We also assume that the relevant standard deviation will be 10%, i. e. that a fr =10 /100 . Therefore, the relevant lower and upper limits for the uniform distribution of fp=D2 tum out to be a fp : ( un~orm_D2 )- ,fj a unitbrnLD2 : 0.327 ( 2 0) { b.fp - ( uniform_D 2 ) + ,fj a unitomLD 2 - 0.673 The next Drake random variable is the number ne of "Earth-type" planets in a given star system. Taking example from the Solar System, since only the Earth is truly "Earth-type", the mean value of ne is clearly I, but the standard deviation is not zero if we assume that Mars also may be regarded as Earth type. Since there are thus two Earth-type planets in the Solar System, we must assume a standard deviation of 1/ ,fj =0.577 to compensate the ,fj appearing in ( 17) in order to finally yield two "Earth type" planets (Earth and Mars) for the upper limit of the random variable ne. In other words, we assume that The next four Drake random variables have even more "arbitrarily" assumed values that we simply assume for the sake of making up a numerical example of our Statistical Drake equation with uniform entry distributions. So, we really make no assumption about the astronomy, or the biology, or the sociology of the Drake equation: we just care about its mathematics. All our assumed entries are given in Table l. Please notice that, had we assumed all the standard deviations to equal zero in Table I, then our Statistical Drake equation (3) would have obviously reduced to the classical Drake equation (1), and the resulting number of civilizations in the Galaxy would have turned out to be 3500: IN =3500 1 . (22) This is the important deterministic number that we will use in the sequel of this paper for comparison with our statistical results on the mean value of N, i.e. (N). This will be explained in Sections 3.3 and 5. UNCLASSIFIED// POil OFFI@IAb W&i 01'11.¥ 34= UNCLASSIFIED/ /FOR OFFIEil.t.k W&i Q~lls¥ 1 s := 3 · o-10 9 s := - s cr s := 1-10 9 ·o 10 μfp := fp fp := 100 crfp := 100 1 ne := 1 μne := ne crne := - ,/3 fl :=~ μfl := fl crt1 = ~ 100 100 fi := 20 μfi := fi crfi := ~ 100 100 fc := 20 - 100 μfc := fc 10 crfc := - 100 10000 fl. :=-- μfl. := fl. 1000 crfl. := -- 1010 101 0 s-fp -ne - fl -fi.fc .fl. - = 3500 Table 1. Input values (i.e. mean values and standard devi ations) for the seven Drake uni fo rm random variables D;. The fi rst column on the left lists the seven input sheer numbers that al so become the mean values (middle co lumn). Finally the last column on the right lists the seven input standard dev iations. The bottom line is the cl assical Drake equation ( 1). 3.2 STEP 6: COMPUTING THE LOGS OF THE 7 UNIFORMY DISTRIB UTED DRAKE RANDOM VARIABLES D; Intuiti vely speakiJ1g, the natural log of a unifonnly disuibuted random vari able may not be another uniforml y distributed random variable! This is obvious fro m the tri vial diagram of y = ln (x) shown belo w: Natural logarithm of x - _,,,,,,,,_,,. ~ r 2 3 4 5 POSITIVE independent variable x Figure 1. The simple function y = ln (x). So, if we have a unifo rml y di stributed random variable D; with lower limit a;and upper limit b;, the random variable Y; =ln(D;) i = l, ...,7 (23) must have its range limited in between the lower limit l11 (a;) and the upper limi t ln(b,). In other words, this are the lower and upper limits of the relevant probability density functio n fr; (y). But what is the actual analytic expression of such a pdf/ . To find it, we must resort to the general transformation law for random variables, defined by equation (9). Here we obviously have y = g(x) = ln(x) (24) That, upon inversion, yields the single root (25) On the other hand, differentiating (24) one gets UNCLASSIFIED//FOR OFFIEil.t.k W&i QNls¥ 35
• • UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ where (25) was already used in the last step. By virtue of the uniform probability density function (I 0) and of (26), the general transformation law (9) finally yields 1n other words, the requested pdf of Y; is eY ,--,---,,-----,-.,.., fr,(y) =-- i = l, ...,7 lm(a;) :,;y:,;in(b;)I (28) b; - a; Probability density functions of the natural logs of all the uniformly distributed Drake random variables D; . This is indeed a positive function of y over the interval !n(a;):,; y:,; ln(b;) , as for every pdf, and it is easy to see that its normalization condition is fulfilled: ... (29) Next we want to find the mean value and standard deviation of Y; , since these play a crucial role for future developments. The mean value (Y;) is given by !n(b;) ( ) s.•n(b;) y,e Y (Y.) = J. Y· fr y dy = -- dy 1 Jn(a,) ' !n(a,)b; -a; b; [in(b; )-1]- a,-[in(a; )-1] = --'-''----''--'--'--=--'-"-"--'-'---= (30) This is thus the mean value of the natural log of all the uniformly distributed Drake random variables D; In order to find the variance also, we must first compute the mean value of the square of Y;, that is ( 2) J.in(b.) 2 { .) J.!n(/J.) y2 • e Y Y; = y ·fry dy= --dy h1(<1;) ' !n(a;) b; - Cl; = b; ~ 2 (b; )- 21n(b; )+ 2] - a; [ln 2 (a; )- 2ln(a; )+ 2) b; - a; ... (32) The variance of Y; = ln(D;) is now given by (32) minus the square of (31) , that, after a few reductions, yield: 2 2 a .b.[ln(b .)- ln(a .)] 2 a - a _ 1_ , , , , (33) r, - In(o,) - (b; _ a;)2 Whence the corresponding standard deviation 1 _ a;b; [in(b;)- in(a;)] 2 (34) (b; - a;)2 Let us now turn to another topic: the use of Fourier transforms, that, in probability theory, are called "characteristic functions," Following again the notations of Papouli s (ref. [5]) we call "characteristic function", <Dy, (s) , of an assigned probability distribution Y; , the Fourier transform of the relevant probability density function, that is (with j = ~) The use of characteristic functions simplifies things greatly. For instance, the calculation of all moments of a known pdf becomes trivial if the relevant characteristic function is known, and greatly simplified also are the proofs of important theorems of statistics, like the Central Limit Theorem that we will use in Section 4. Another important result is that the characteristic function of the sum of a finite number of independent random variables is simply given by the product of the corresponding characteristic functions. This is just the case we are facing in the Statistical Drake equation (3) and so we are now led to find the characteristic function of the random variable Y; , i.e. UNCLASSIFIED/ , 'FOA OFlilCI0 I. P!SE ON! X 36UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ (36) Thus, the characteristic function of the natural log ofthe Drake uniform random variable D; is given by 3.3 STEP 7: FINDING THE PROBABILITY DENSITY FUNCTION OF N, BUT ONLY NUMERICALLY NOT ANALYTICALLY Having found the characteristic functions <l>y/s) of the logs of the seven input random variables D; , we can now immediately find the characteristic fu □ ctio □ of the random variable Y = ln(N) defined by (5). 1n fact, by virtue of (4), of the well-known Fourier transform property stating that "the Fourier transform of a convolution is the product of the Fourier transforms", and of (37), it immediately follows that <l>y(() equals the product of the seven Cf> y 1 (s): The next step is to invert this Fourier transform in order to get the probability density function of the random variable Y = ln(N). In other words, we must compute the following inverse Fourier transform -oo oo [ 7 bl +)( i+ j( l = - 1 fe -i(y IT ;_ - a;_ dc;'.(39) 2n - co i= I (b; a;)(l+ Js) This author regrets that he was unable to compute the last integral analytically. He had to compute it numerically for the particular values of the 14 a; and b; that follow from Table 1 and equations 17. The result was the probability density function for Y = ln(N) plotted in the following Figure 2. ... >- PROB. DENSITY FUNCTION OF Y=ln(N) 0 0.4 C 0 ·.:::, ..2 Q 0.3 -f 0.2 ii "' ~ 0.1 :z; ..8 • ~I....- , / ' I \ \ O O I 2 3 4 5 6 7 8 9 10 11 12 e ll, Independent variable Y = ln(N) Figure 2. Probability density function of Y = ln(N) computed numericaJly by virtue of the integral (39). The two "funny gaps" in the curve are due to the numeric limitations in the MathCad numeric solver that the author used for this numeric computation. We are now just one more step from finding the probability density of N, the number of ExtraTerrestrial Civilizations in the Galaxy predicted by our Statistical Drake equation (3) . The point here is to transfer from the probability density function of Y to that of N, knowing that Y = ln(N) , or alternatively, that N=exp(Y), as stated by (6). We must thus resort to the transformation law of random variables (9) by setting (40) This, upon inversion, yields the single root (41) On the other hand, differentiating (40) one gets (42) where (41) was already used in the last step. The general transformation law (9) finally yields UNCLASSIFIED/ /EAR AFFJCJA 1 1 !SF QN 1 Y 37
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ This probability density function f N (y) was computed numerically by using (43) and the numeric curve given by (39), and the result is shown in Figure 3. Z 4 ·l 0 -4PROBA BILITY DENSITY FUNCTIO OF N ..... 0 .g 3·10-4 {) C: ..: 2·10-4 ·i ~ 1 ·10-4 1000 2000 3000 4000 N = umber of ET Civilizations in Galaxy Figure 3. The numeric (and not analytic) probability density function curve f N (y) of the number N of ExtraTerrestrial Civilizations in the Galaxy according to the Statistical Drake equation (3). We see that the curve peak (i.e. the mode) is very close to low values of N, but the tail on the right is high, meaning that the resulting mean value ( N) is of the order of thousands. We now want to compute the mean value (N) of the probability density (43). Clearly, it is given by (N) = f 00 y J,v(y)dy. (44) 0 This integral too was computed numerically, and the result was a perfect match with N=3500 of (22), that is (N) = 3499.99880 177509 + 0.OOCXX:XH2 4914686i (45) Note that this result was computed numerically in the complex domain because of the Fourier transforms, and that the real part is virtually 3500 (as expected) while the imaginary part is virtually zero because of the rounding errors. So, this result is excellent, and proves that the theory presented so far is mathematically correct. Finally we want to consider the standard deviation. This also had to be computed numerically, resulting in uN =3953.42910 143389 +o.cx: x:x:x: x:m 28CXXJ58i . (46) This standard deviation, higher than the mean value, implies that N might range in between 0 and 7453. This completes our study of the probability density function of N if the seven uniform Drake input random variable D; have the mean values and standard deviations listed in Table I . We conclude that, unfortunately, even under the simplifying assumptions that the D; be 11nifor111ly distributed, it is impossible to solve the full problem a11alytically, since all calculations beyond equation (38) had to be performed numerically. This is no good. Shall we thus loose faith, and declare "impossible" the task of finding an analytic expression for the probability density function f N (y) ? Rather surprisingly, the answer is "no", and there is indeed a way out of this dead-end, as we shall see in the next section. 5. THE CENTRAL LIMIT THEOREM (CLT) OF STATISTICS Indeed there is a good, approximating analytical expression for ftv (y) , and this is the following lognormal probability density Junction To understand why, we must resort to what is perhaps the most beautiful theorem of Statistics : the Central Limit Theorem (abbreviated CLT). Historically, the CLT was in fact proven first in 190 I by the Russian mathematician Alexandr Lyapunov (1857-1918), and later (1920) by the Finnish mathematician Jar! Waldemar Lindeberg (1876-1932) under weaker conditions. These conditions are certainly fulfilled in the context of the Drake equation because of the "reality" of the astronomy, biology and sociology involved with it, and we are not going to discuss this point any further here. A good, synthetic description of the Central Limit Theorem (CLT) of Statistics is found at the Wikipedia site (ref. (7)) to which the reader is refetTed for more details, such as the equations for the Lyapunov and the Lindeberg conditions, making the theorem "rigorously" valid. UNCLASSIFIED/ J 'FQA QFFICiIPL. !Iii 011! X 38UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Put in loose terms, the CLT states that, if one has a sum of random variables eve11 NOT identically distributed, this sum tends to a normal distribution whe11 the 11umber of tenns maki11g up the sum te11ds to infinity. Also, the normal distribution mean value is the sum of the mean values of the addend random variables, and the normal distribution variance is the sum of the variances of the addend random variables. Let us now write down the equations of the CLT in the form needed to apply it to our Statistical Drake equation (3). The idea is to apply the CLT to the sum of random variables given by (4) and (5) whatever their probability distributions ca11 possibly be. Tn other words, the CLT applied to the Statistical Drake equation (3) leads immediately to the following three equations: I) The sum of the (arbitrarily distributed) independent random variables Y; makes up the new random variable Y. 2) The sum of their mean values makes up the new mean value of Y. 3) The sum of their variances makes up the new variance of Y. In equations: 7 y = :~::vi i =I 7 (Y) = I(Y; ) (48) i = I 7 2 = "' O'y L..Jo-r 2 , i= I This completes our synthetic description of the CLT for sw11s of random variables . 6. THE LOGNORMAL DISTRIBTIO IS THE DISTRIBUTION OF THE NUMBER N OF EXT RATERRESTRIAL CIVILIZATIONS IN THE GALAXY The CLT may of course be exte11ded to products of random variables upon taki11g the logs of both sides, just as we did in equatio11 (3 ). It then follows that the exponent random variable, like Y i11 (6), tends to a 11ormal random variable, and, as a co 11seque 11ce, it follows that the base random variable, like N i11 (6), te11ds to a lognormal random variable. To understand this fact better in mathematical terms consider again of the transformation law (9) of random variables. The question is: what is the probability density function of the random variable N in equation (6), that is, what is the probability density function of the lognormal distribution? To find it, set (49) This, upon inversion, yields the single root x 1 (y) =x(y) =!n(y). (50) On the other hand, differentiating (49) one gets where (50) was already used in the last step. The general transformation law (9) finally yields ( ) "' fx(x;(y)) 1 ( ( )) fNY= L..J i '( ()~=- /ylny • (52) i g X; )' ~ 11 y Therefore, replacing the probability density on the right by virtue of the well-known normal (or Gaussian) distribution given by equation (7), the lognormal distribution of equation (47) is found , and the derivation of the lognormal distribution from the normal distribution is proved. In view of future calculations, it is also useful to point out the so-called "Gaussian integral", that is: B2 f"' e - Ax 2 e 8 ·x dx = ~ • e 4 A A> 0 B = real. (53) L ., VA ' ' This follows immediately from the normalization condition of the Gaussian (7), that is oo (x- pf 1 -- , - J-- e lu· dx=l (54) &a- ' --Oj just upon expanding the square at the exponent and making the two replacements (we skip all steps) l A=-- >0, 2 2 a - (55) l B = ; 2 = real. UNCLASSIFIED/ /FOR 8FFIGl.t.k W&i ,n11.¥ 39
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ In the sequel of this paper we shall denote the independent variable of the lognormal distribution (47) by a lower case letter n to remind the reader that corresponding random variable N is the positive integer number of ExtraTerrestriaJ Civilizations in the Galaxy. In other words, n will be treated as a positive real number in all calculations to follow because it is a "large" number (i.e. a continuous variable) compared to the only civilization that we know of, i.e. ourselves. In conclusion, from 110w 011 the log11or111al probability density f11nction of N will be written as (1n(11}--μ)2 1 1 -~ ( ) (56) n v2na IN n =-· r;:;- e (n~ O) Having so said, we now turn to the statistical properties of the lognormal distribution (55), i.e. to the statistical properties that describe the number N of ExtraTerrestrial Civilizations in the Galaxy. Our first goal is to prove an equation yielding all the moments of the lognormal distribution (56), that is, for every non-negative integer k = 0, I, 2, . . . one has (57) The relevant proof starts with the definition of the k d1 moment (111[11]-p)' k I I -~ = r n • ~ • .Ji; a • e dn 0 One then transforms the above integral by virtue of the substitution In[n]= z. (58) The new integral in z is then seen to reduce to the Gaussian integral (53) (we skip all steps here) and (57) follows Upon setting k = 0 into (56), the normalization condition for f N (n) follows Upon setting k =1 into (56), the important mean value of the ra11dom variable N is found (60) Upon setting k = 2 into (56), the mean value of the square of the random variable N is found /N2) - 2 11 2a 2 \ -e e . (61) The variance of N now follows from the last two formulae: (62) The square root of this is the important standard deviation f onnula for the N random variable (63) The third moment is obtained upon setting k = 3 into (56) (64) Finally, upon setting k =4, the fourth moment of N is found (65) Our next goal is to find the cumulants of N. In principle, we could compute all the cumulants K; from the generic i-th moment p; by virtue of the recursion formula (see ref. [8]) • i- 1 (i - 1) . (66) K; =A- L k-1 K k f.l11-k· k=i UNCLASSIFIED/ /rOR orrl@IAL HSE O,.LY 40
UNCLASSIFIED/ /fOR OFFI&IAk WliF QPU,¥ In practice, however, here we shall confine ourselves to the computation of the first four cumulants only because they only are required to find the skewness and kurtosis of the distribution. Then, the first four cumulants in terms of the first four moments read: These equations yield, respectively: (68) (69) (70) From these we derive the skewness (72) and the kurtosis (73) Finally, we want to find the mode of the lognormal probability density function, i.e. the abscissa of its peak. To do so, we must first compute the derivative of the probability density function f N (n) of equation (56), and then set it equal to zero. This derivative is actually the derivative of the ratio of two functions of n, as it plainly appears from (57). Thus, let us set for a moment E(n) = (1n [n] - ,u)2 (74) 2a 2 where "E" stands for "exponent," Upon differentiating this, one gets 1 E' (n) = - - •2(In [n] - ,u)· _!_ . (75) 2a 2 n But the lognormal probability density function (56), by virtue of (74), now reads (76) So that its derivative is dfET_Dis1ana,' (r) - e- E(" )E' (n )· n -1 · e- E(") dr .Ji;a • n2 I -e- E(")[E'(n)·n + i] (77) = &a. n2 Setting this derivative equal to zero means setting E '(n) · n+l=O (78) That is, upon replacing (75), ~ -(In[n]- μ)+1 =0. (79) a Rean-anging, this becomes In[n]- μ+a 2 = 0 (80) and finally I _ I _ fl - a ' 11 ntJde = 11pcak - e e (81) This is the most likely number of ExtraTerrestrial Civilizations i11 the Galaxy. How likely? To find the value of the probability density function f N (n) corresponding to this value of the mode, we must obviously replace (81) into (56). After a few rearrangements, one then gets UNCLASSIFIED/ ; 'FOA OlililCil0 L. P!SF ON! Y 41UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ (82) 2 f x ' erf(x)= r e-:-dz (85) This is "how likely" the most likely number of ExtraTerrestrial Civilizations in the Galaxy is, i.e. it is the peak height in the lognormal probability density function f N(n) . Next to the mode, the median m (ref. (9)) is one more stati stical number used to characterize any probability distribution. It is defined as the independent variable abscissa m such that a realization of the random variable will take up a value lower than m with 50% probability or a value higher than m with 50% probability again. In other words, the median m splits up our probability density in exactly two equally probable parts. Since the probability of occurrence of the random event equals the area under its density curve (i.e. the definite integral under its density curve) then the median m (of the lognormal distribution, in this case) is defined as the integral upper limit m: (111(11 )-,,1 )2 f 111 JN(n)dn = f "'_!__ rd- e---;;;;r- == (83) Jo Jo n v 2na 2 In order to find m , we may not differen tiate (83) with respect to m , since the "precise" factor ½ on the right would then disappear into a zero. On the contrary, we may try to perform the obvious substitution zzO (84) into the integral (83) to reduce it to the following integral defining the error function erf(z) -,/ Ji 0 Then, after a few reductions that we skip for the sake of brevity, the full equation (83) is turned into (86) that is e,f[ln(m) - p) == 0 (87) ✓ 2a Since from the definition (85) one obviously has erf(0)=0, (87) becomes ln(m)- μ == 0 (88) ✓ 2a whence finally I rredian = m = e" I. (89) This is the median of the lognormal distribution of N. In other words, this is the number of ExtraTerrestrial civilizations in the Galaxy such that, with 50% probability the actual value of N will be lower than this median, and with 50% probability it will be higher. ln conclusion, we feel useful to summarize all the equations that we derived about the random variable N in the fo llowing Table 2. Random variable Probability distribution Probability density function N =number of communicating ET civilizations in Galaxy Lognormal (1 11(11}--p)' 1 1 ---;;;;r - fN(n) = - · ..n,; e (n z O) n 2na Mean value (T2 - (N) = e" e 2 Variance a 1 = e 2 " ea2 (ea2 -1) Standard deviation a' a N = e" e2 .Jea' -1 UNCLASSIFIED/ /fOR OFfl&IAk Wlii QNls¥ 42
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ All the moments, i.e. k-th moment Mode(= abscissa of the lognormal peak) Value of the Mode Peak Median(= fifty-fifty probability value for N) Skewness Kurtosis Expression of μ in terms of the lower (a;) and upper (b;) limits of the Drake uniform input random variables D; Expression of a 2 in terms of the lower (a;) and upper (b;) limits of the Drake uniform input random variables D; - - JI -a2 nnnde = npeak - e e a l I - 11 , f ( ) N nTIDde = ~ • e • e - -,121r a Table 2. Summary of the properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy . We want to complete this section about the lognormal probability density function (56) by finding out its numeric values for the inputs to the Statistical Drake equation (3) listed in Table I. According to the CLT, the mean value μ to be inserted into the lognormal density (56) is given (according to the second equation (48)) by the sum of all the mean values (Y;), that is, by virtue of (31 ), by: Upon replacing the 14 a; and b; listed in Table 1 into (90), the following 11umeric mea11 value μ is found 1μ,,,7.462116 1 (91) Similarly, to get the numeric variance a 2 one must resort to the last of equations (48) and to (33): (92) a 2 yielding the following numeric varia11ce to be inserted into the lognormal pdf (56) la 2 ,,, 1.938725 1 (93) whence the numeric standard deviation a 1 a ,,, 1.392381 t- (94) Upon replacing these two numeric values (84) and (86) into the lognormal pdf (56), the latter is perfectly determined. It is plotted in Figure 4 hereafter as the thin curve. In other words, Figure 4 shows the lognormal distribution for the number N of ExtraTerrestrial Civilizations in the Galaxy derived from the Central Limit Theorem as applied to the Drake equation (with the input data listed i11 Table 1). We now like to point out the most important statistical properties of this lognormal pdf: 1) Mean Value of N. This is given by equation (60) with μ and a given by (91) and (94), respectively: UNCLASSIFIED/ /FOR OFFICIO! !!SF ON! X 43
UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ (N) = eμ e 2 "'4589 .559 . (95) In other words, there are 4590 ET Civilizatio11s in the Galaxy according the Central Limit Theorem of Statistics with the inputs of Table 1. This ,mmber 4590 is HIGHER than the 3500 foreseen by the classical Drake equation working with sheer numbers only, rather than with probability distributions. Thus equation (95) IS GOOD FOR NEWS FOR SETI, since it shows that the expected number of ETs is HIGHER with an adequate statistical treatment than just with the too simple Drake sheer numbers of (1). 2) Variance of N. The variance of the Iognormal distribution is given by (62) and turns out to be a huge number: 3) Standard deviation of N. The standard deviation of the lognormal distribution is given by (63) and turns out to be: (97) Again, this is GOOD NEWS FOR SETI. In fact, such a high standard deviation means that N may range from very low values (zero, theoretically, and one since Humanity exists) up to tens of thousands (4590+11195=15785 is (95)+(97)). 4) Mode of N. The mode (= peak abscissa) of the lognormal distribution of N is given by (81 ), and has a surprisingly low numeric value: This is well shown in Figure 4: the mode peak is very pronounced and close to the origin, but the right tail is high, and this means that the mean value of the distribution is much higher than the mode: 4590»250. 5) Median of N. The median (= fifty -fifty abscissa, splitting the pdf in two exactly equi-probable parts) of the lognormal distribution of N is given by (89), and has the numeric value: (99) Tn words, assuming the input values listed in Table 1, we have exactly a 50% probability that the actual value of N is lower than 1740, and 50% that it is higher than 1740. 7. COMPARING THE CLT RESULTS WITH THE NON-CLT RESULTS The time is now ripe to compare the CLT based results about the lognormal distribution of N, just described in Section 5, against the Non-CLT based results obtained numerically in Section 3.3 To do so in a simple, visual way, let us plot on the same diagram two curves: 1) The numeric curves appearing in Figure 2 and obtained after laborious Fourier transform calculations in the complex domain, and 2) The lognormal distribution (56) with numeric μ and a given by (91) and (94) respectively. We see that the two curves are virtually coincident for values of N larger than 1500. This is a consequence of the law of large numbers, of which the CLT is just one ofthe 111a11y facets. Similarly it happens for natural log of N, i.e. the random variable Y of (5), that is plotted in Figure 5 both in its normal curve version (thin curve) and in its numeric version, obtained via Fourier transforms and already shown in Figure 2. The conclusion is simple: from now 011 we shall discard forever the 1111meric calc11lations a11d we'll stick only to the equations derived by virtue of the CLT, i.e. to the lognormal (56) and its co11seque11ces. UNCLASSIFIED//509 OFFICIO! 11SF QN 1X 44
UNCLASSIFIED/ /fOR OFFI&IAk WliF QPU,¥ 6 ·10-4 1000 2000 3000 4000 N = Number of ET Civilizations in Galaxy Figure 4. Comparing the two probability density functions of the random variable N found: 1) At the end of Section 3.3. in a purely numeric way and without resorting to the CLT at all (thick curve) and 2) AnalyticaJly by using the CLT and the relevant lognormal approximation (thin curve). PROBABILITY DENSITY FUNCTION OF Y=ln(N) 0.5 >- 0.4 ..... 0 g .B § 0.3 ..... • ~ 5 "O 0.2 _ q ] " .D e 0.1 0.. 00 2 3 12 Independent variable Y = ln(N ) Figure 5. Comparing the two probability density functions of the random variable Y=ln(N) found: I) At the end of Section 3.3 . in a purely numeric way and without resorting to the CLTat all (thick curve) and 2) Analytically by using the CLT and the relevant normal (Gaussian) approximation (thin Gauss ian curve). 8. DISTANCE OF THE NEAREST and in several web sites, the solution to this EXTRA TERRESTRIAL CIVILIZATION problem is reported with only slight differences in AS A PROBABILITY DISTRIBUTION the mathematical proofs among the various authors . In the first of the coming two sections (section 7 .1 ) As an application of the Statistical Drake we derive the expression for this "ET_Distance" Equation developed in the previous sections of this (as we like to denote it) in the classical, non paper, we now want to consider the problem of probabilistic way: in other words, this is the estimating the distance of the ExtraTerrestrial classical, deterministic derivation. In the second Civilization nearest to us in the Galaxy. In all section (7.2) we provide the probabilistic Astrobiology textbooks (see, for instance, ref. [10)) derivation, arising from our Statistical Drake UNCLASSIFIED/}FOA OlililCil0L. 1155 ON!¥ 45
UNCLASSIFIED/ /fOR OFFI&Ial.k Wlili QPU,¥ Equation, of the corresponding probability density function fET_Di stance(r) : here r is the distance between us and the nearest ET civilization assumed as the independent variable of its own prob~bility density function. The ensuing sections provide more mathematical details about this .firr_oi stana, (r) such as its mean value, variance, standard deviation, all central moments, mode, median, cumulants, skewness and kurtosis . CLASSICAL, NON-PROBABILISTIC DERIVATION OF THE DISTANCE OF THE NEAREST ET CIVILIZATION Consider the Galactic Disk and assume that: 1) The diameter of the Galaxy is (about) 100,000 light years, (abbreviated ly) i.e. its radius, R calaxy• is about 50,000 ly. 2) The thickness of the Galactic Disk at half-way from its center, hcalaxy , is about 16,000 ly. Then 3) The volume of the Galaxy may be approximated as the volume of the corresponding cylinder, i.e. V - R 1 h (100) Gal«ry - 1l Ga/,uy 4) Now consider the sphere around us having a radius r. The volume of such as sphere is 4 (Er Distance) 3 VOw·_ Sphere = 3 1l' 2 (101) In the last equation, we had to divide the distance "ET_Distance" between ourselves and the nearest ET Civilization by 2 because we are now going to make the unwarranted assumption that all ET Civilizations are equally space from each other in the Galaxy! This is a crazy assumption, clearly, and should be replaced by more scientifically grounded assumptions as soon as we know more about our Galactic Neighbourhood. At the moment, however, this is the best guess that we can make, and so we shall take it for granted, although we are aware that this is weak point in the reasoning. Having thus assumed that ET Civilizations are UNIFORMLY SPACED IN THE GALAXY ' we can write down this proportion: Vcal<Lry = Vo11r_ Sphere (102) N That is, upon replacing both (100) and (101) into (102): 4 (Er Distance ) 3 1l RJalaxy h = 3" ---=- 2- - (103) N I The only unknown in the last equation is ET_Distance, and so we may solve for it, thus getting the: (A VERA GE) DISTANCE BETWEEN ANY PAIR OF NEIGHBOURING CIVILIZATIONS IN THE GALAXY 116 R 2 h _ V Galaxy C Er D . (104) istance - 1 ,.--; = - - tN ifii where the positive constant C is defined by C =3 6 Ria1a.,y h calcuy ""28845 light years . ( l 05) Equations (104) and (105) are the starting point for our first application of the Statistical Drake equation, that we discuss in detail in the coming sections of this paper. PROBABILISTIC DERIVATION OF THE PROBABILITY DE SITY FUNCTION FOR ET_DISTA CE ~he probability density function (pdt) yielding the distance of the ET Civilization nearest to us in the Galaxy and pre ented in this section, was discovered by this author on September 5th' 2007. He did not disclose it to other scientists until the SETI meeting run by the famous mathematical physicist and popular science author, Paul Davies, at the "Beyond" Center of the University of Arizona at Phoenix, on February 5-6-7-8, 2008. This meeting was also attended by SETI Institute experts Jill Tarter, Seth Shostak, Doug Vakoch, Tom Pierson and others. During this author's talk, Paul Davies suggested to call "the Maccone distribution" the new probability density function that yields the ET_Distance and is derived in this section. UNCLASSIFIED/ ffOR OFFI&Ial.L Wlili QPII..¥ 46
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Let us go back to equation (104). Since N is now a random variable (obeying the lognorma1 distribution), it fo ll ows that the ET_Distance must be a random variable as well. Hence it must have some unknown probability density function that we denote by (106) f ET_Di stana, (r) where r is the new independent variable of such a probability distribution (it is denoted by r to remind the reader that it expresses the three dimensional radial distance separating us from the nearest ET civilization in a full spherical symmetry of the space around us). The question then is: what is the unknown probability distribution (106) of the ET_Distance? We can answer this question upon making the two formal substitutions N ➔ x { (107) Er _distance ➔ y into the transformation law (8) for random variables. As a consequence, ( I04) takes form I C - y = g(x) = -,- = C • x 3 . (108) v;; In order to find the unknown probability density f ET_Di stru,., (r) , we now to apply the rule (9) to (108). First, notice that (108), when inverted to yield the various roots x; (y), yields a single real root only (109) Then, the summation in (9) reduces to one term only. Second, differentiating (108) one finds 4 C - - g ' (x) =-- ·x 3 . (110) 3 Thus, the relevant absolute value reads (111) Upon replacing (111) into (9), we then find This is the denominator of (9) . The numerator simply is the lognormal probability density function (56) where the old independent variable x must now be re-written in terms of the new independent variable y by virtue of (109). By doing so, we finally an-ive at the new probability density function f y (y) Rearranging and replacing y by r, the final form is: 2 ( In[~]- 11] 3 1 - 2u' f ET_d istana, (r) = - • ~ • e (113) 2 r 'I/ L.Jr a- Now, just replace C in ( 113) by virtue of ( 105). Then: We have discovered the probability density function yielding the probability of finding the nearest ExtraTerrestrial Civilization in the Galaxy in the spherical shell between the distances r and r+dr from Earth: (,{ 6 Ri.t11 a~~ hc 0 1a.Q}l'r 3 l 2 0- l fET_Di stanaJr)=-· ~ ·e 2 r "1.,1r a (114) holding for r .:: 0 . STATISTICAL PROPERTIES OF THIS DISTRIBUTION UNCLASSIFIED/ /FOR &FFIGIAk lalii ODIL:¥ 47UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ We now want to study this probability distribution in detail. Our next questions are: l) What is its mean value? 2) What are its variance and standard deviation? 3) What are its moments to any higher order? 4) What are its cumulants? 5) What are its skewness and kurtosis? 6) What are the coordinates of its peak, i.e. the mode (peak abscissa) and its ordinate? 7) What is its median? The first three points in the list are all covered by the following theorem: all the moments of ( 113) are given by (here k is the generic and non negative integer exponent, i.e. k = 0, I, 2, 3,... ~ 0) (Er_Distance*) = f'/ · f ET_Distana,(r)dr ( 1n[~ ]-μr dr =fr*-r~(T -e (115) To prove this result, one first transforms the above integral by virtue of the substitution (116) Then the new integral in z is then seen to reduce to the known Gaussian integral (53) and, after several reductions that we skip for the sake of brevity, (115) follows from (53). In other words, we have proven that μ 2 (J2 (Er_Distancek) = Ck e - k3 / • 18 . (117) Upon setting k = 0 into (117), the normalization condition for f ET_Distan"' (r) follows i f ET Distam e(r)dr = l . (118) 0 - Upon setting k = l into (117), the important mean val11e of the random variable ET_Distance isfo1111d f.-l a z (Ef_Distance) =C e 3 e 18 . (119) Upon setting k = 2 into (l 17), the mean value of the square of the random variable ET_Distance is found 2 2 2 - p - (1 ( Ef_Distance 2 ) = C2 e 3 e 9 (120) The variance of ET _Distance now follows from the last two formulae with a few reductions: CT~T_Distan"' = (Ef_Distance 2 )- (Ef_Distance) 2 (121) So, the variance ofET_Distance is (122) The square root of this is the important standard deviation of the ET_Distance random variable _f!_ (11 ,-;i- - (TET _Di S (ana, =C e 3 e• 8 ~e9 -1 . (123) The third moment is obtained upon setting k = 3 into ( 117) (1 2 (Er_Distance 3 ) = C 3 e-p e 2 (124) Finally, upon setting k = 4 into ( 117), the fourth moment of ET_Distance is found 4 8 l (Er_Distance 4 )=C4 e 3 -μ e 9(1 (125) UNCLASSIFIED//POR 9Pfl@IAL H§E 8PUs:¥ 48
3 UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ a 2 5,,. 2 a '] Our next goal is to find the cumulants of the e- P [ e 2 - 3 e 18 + 2 e 6 ET_Di stance. 1n principle, we could compute all the cumulants K; from the generic i-th moment μ; by virtue of the recursion formula (see ref. [8]) ' i- 1 ( i -1) ' (126) k=i k - 1 K; = JI; -z: Kk J.111-k · ln practice, however, here we shall confine ourselves to the computation of the fust four cumulants because they only are required to find the skewness and kurtosis of the distribution ( 113). Then, the first four cumulants in terms of the first four moments read: These equations yield, respectively: _!!_ a 2 K .=Ce3el8_ (128) (129) (130) Sa ' 5a 2 4a 2 a 2 2a 2 ]2 C 3 [ e- 9 - -4e 9 -3e 9 +12 e 3 -6e_ 9 _ ... (132) and the kurtosis 4 0"2 2 a 2 K - (K 4 )2 =e 9 +2e 3 +3e 9 -6 . (133) 2 Next we want to find the mode of this distribution, i.e. the abscissa of its peak. To do so, we must fust compute the derivative of the probability density function fET_Oistame(r) of (113), and then set it equal to zero. This derivative is actually the derivative of the ratio of two functions of r, as its plainly appears from (113). Thus, let us set for a moment (134) where "E" stands for "exponent," Upon differentiating, one gets (13 1) = - 1 •[ln[£.] - μ)·(- 3).!.. (135) c,2 ,-3 ,. 2 4a 2 4p [ sa 5a2 a ' 2a'] = C 4 e - 3 e- 9 - -4 e- 9 - - 3e_9 _ + 12 e 3 - 6e_9 _ But the probability density function (113) now reads 3 e-E(r) From these we derive the skewness fET Di stana:(r)= ~ ·-- (136) - ..,;2,rc, r So that its derivative is dfET_Di siana:, (r) 3 -e - E(r) E' (r)· r-1 · e- E(r) dr = &a. ,.2 UNCLASSIFIED// POil OFFI@IAb W&I! 91'11.¥ 49
UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ 3 -e-£(,-) [E '(,}r+l] (137) .Ji;a . ,.2 Setting this derivative equal to zero means setting E'(r)·r+l=O (138) That is, upon replacing (l 35) into (138), we get Rearranging, this becomes that is (141) whence In[£] = !!_+ i!_ (142) r 3 9 and finally _!:!_ 0'"2 r;mde"',.peak=Ce 3e 9 (143) This is the most likely ET_Distance/rom Earth. How likely? To find the value of the probability density function f ET_Disim,"' (r) corresponding to this value of the mode, we must obviously replace () into (). After a few rearrangements, which we skip for the sake of brevity, one gets Peak Value of fET_Distana,(r) = fET_Distana,(1rroctcl ... (144) This is the peak height in the pdf f ET_Dist"""' (r) • Next to the mode, the median m (ref. [9]) is one more statistical number used to characterize any probability distribution. It is defined as the independent variable abscissa m such that a realization of the random variable will take up a value lower than m with 50% probability or a value higher than m with 50% probability again. In other words, the median m splits up our probability density in exactly two equally probable parts. Since the probability of occurrence of the random event equals the area under its density curve (i.e. the definite integral under its density curve) then the median m (of the lognormal distribution, in this case) is defined as the integral upper limit m: J, "' I (145) fET Distanu,(r)dr= - 0 - 2 Upon replacing ( 113), this becomes (146) 2 In order to find m, we may not differentiate ( 146) with respect to m, since the "precise" factor ½ on the right would then disappear into a zero. On the contrary, we may try to perform the obvious substitution (147) into the integral (146) to reduce it to the following integral (85) defining the error function erf(z). Then, after a few reductions that we leave to the reader as an exercise, the full equation (145), defining the median, is turned into the corresponding equation involving the error function erj{x) as defined by (85): UNCLASSIFIED/ /FOR OFFI&IAk Wlii ,un.¥ 50UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Random variable Probability distribution Probability density function ET_Distance between any two neighboring ET Civilizations in Galaxy assuming they are UNIFORMLY distributed throughout the whole Galaxy volume. Unnamed (Paul Davies suggested "Maccone distribution") ( in[ 6 ~""';i''G• 1 •"t]- 11J 3 1 2 cr 2 f ET Di stana,(r) = - • .{i; • e - r 21t O" (Defining the positive numeric constant C) C = V6 /iJ,a/a.,y hGa/a.,y ~ 28845 light years Mean value μ <,2 (Er_Distance) = C e- 3 e 18 Variance 2 - 2 -3 2 ;, 9 <,2 [ 9 <,2 l O" ET_Distana: - C e e e - 1 Standard deviation _f!_U2["S2 - 3 18 9 _ O" ET_Distano, - Ce e e I All the moments, i.e. k-th moment ,, a 2 - kl!. k·· - (Er_Distancek) = C k e 3 e 18 Mode(= abscissa of the probability density function peak) _!!_ u' - - 3 e 9 r ITDde = rpeak = Ce Value of the Mode Peak Peak Value of fET_Di stru,.,(r) = <,2 1!. - 3 · e 3 • e 18 = f ET Distanre ( ,-rmdc) = -& - C 21r O" Median (= fifty-fifty probability value for ET Distance) _!:!. 3 rredian = m = Ce l Skewness 2 2 2 u 5a a [ e-μe 2 - 3e· 8 +2e 6 _!S_ = 3 3 (K4 }2 8a 2 5a 1 2 4 u 2 o - 2u 2 ]2 C 3 e- 9 - -4e _ _ [ 9 -3e_ _ 9 +12 e 3 -6e _ _ 9 Kurtosis 4 0"2 <, 2 2 0"2 - - - K4 9 9 + 2e 3 +3e -6 (Ki)2 = e Expression of pin terms of the lower (a;) and upper (b;) limits of the Drake uniform input random variables D; J.l= I(Y;)= Ib;[ln(b;) - 1] - a;[In(a;) - 1] i= I i= I b; - a; Expression of 0" 2 in terms of the lower (a;) and upper (b;) limits of the Drake uniform input random variables D; 7 7 a;b;[In(b; ) - In(a; )] 2 O" 2 = I 2 O"y, = I I - (b; - a; ) 2 i=i i= i Table 3. Summary of the properties of the probability distribution that applies to the random variable ET _Distance yielding the (average) distance between any two neighboring communicating civilizations in the Galaxy. UNCLASSIFIED/ /FOA OlililCI0 I. P!SE ON! X 51UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ that is (148) (149) Since from the definition (147) one obviously has erf(0)=0, (149) yields This is the median of the lognormal distribution of N. Ill other words, this is the number of ExtraTerrestrial civilizatio11s i11 the Galaxy such that, with 50% probability the actual value of N will be lower tha11 this median, and with 50% probability it will be higher. In conclusion, we feel useful to summarize all the equations that we derived about the random variable N in the following Table 2. NUMERICAL EXAMPLE OF THE ET_DISTANCE DISTRIBUTION ln this section we provide a numerical example of the analytic calculations carried on so far. Consider the Drake Equation values reported (150) in Table 1. Then, the graph of the corresponding probability density function of the nearest whence finally ET_Distance, f ET_Distan"' (r), is shown in Figure 6. I median= m= Ce-~ 1. (151) DISTANCE OF NEAREST Er_CIVILIZA TION 5.63·10-20 500 I000 I500 2000 2500 3000 3500 4000 4500 5000 ET_ Distance from Earth (ligJ,t years) Figure 6. This is the probability of finding the nearest ExtraTerrestrial Civilization at the distance r from Earth (in light years) if the values assumed in the Drake Equation are those shown in Table I. The relevant probability density function f ET_Distano:Cr) is given by equation (113). Its mode (peak abscissa) equals 1933 light years, but its mean value is higher since the curve has a high tail on the right: the mean value equals in UNCLASSIFIED/fFQA QFFICiIPL. PP&i ODI! Y 52
UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR SETI, inasmuch as the nearest ET Civilization might lie at just 1 sigma = 2670-1309 = 1361 light years from us. From Figure 6, we see that the probability of finding ExtraTenestrials is practically zero up to a distance of about 500 light years from Earth. Then it starts increasing with the increasing distance from Earth, and reaches its maximum at _fl_ ,? - rrmde = rpeak = Ce 3 e 9 ~1933 light years. (152) This is the MOST LIKELY VALUE of the distance at which we can expect to fi11d the nearest ExtraTerrestrial civilization. It is not, however, the mean value of the probability distribution (113) for .feT_Distana,(r) . In fact, the probability density (] l 3) has an infinite tail on the right, as clearly shown in Figure 6, and hence its mean value must be higher than its peak value. As given by ( 119), its mean value is _!!_ 0"2 r,11ea 11 l'alue =C e 3 e 18 ~ 2670 light years. (153) This is the MEAN (value of the) DISTANCE at which we can expect to find ExtraTerrestrials. After having found the above two distances (1933 and 2670 light years, respectively) , the next natural question that arises is: "what is the range, forth and back around the mean value of the distance, within which we can expect to find ExtraTenestrials with "the highest hopes ?," The answer to this question is given by the notion of standard deviation, that we already found to be given by (123) _fl_ 0"2 ~ (TET_Dislana, = Ce 3 e 18 Ve 9 - 1 ~1309 light years. ... (154) More precisely, this is the so called I-sigma (distance) level. Probability theory then shows that the nearest ExtraTenestrial civilization is expected to be located within this range, i.e. within the two distances of (2670-1309) = 1361 light years and (2670+ 1309) = 3979 light years, with probability given by the integral of fET_Distana, (r) taken in between these two lower and upper limits, that is: i 39791ightyears /ET Distana,(r)dr~0.75 = 75% (155) 136 llightyears - In plain words: with 75% probability, the nearest ExtraTerrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by Table 1. If we change those input values, then all the numbers change again. 9. THE "DATA ENRICHMENT PRINCIPLE" AS THE BEST CL T CONSEQUENCE UPON THE STATISTICAL DRAKE EQUATION (ANY NUMBER OF FACTORS ALLOWED) As a fitting climax to all the statistical equations developed so far, let us now state our "DATA ENRICHMENT PRINCIPLE," It simply states that "The Higher the Number of Factors in the Statistical Drake equation, The Better," Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the nonnal distribution when the number of terms in the sum (4) approaches infinity. And this is the case, indeed. However, our "Data Enrichment Principle" has more profound methodological consequences that we cannot explain now, but hope to describe more precisely in one or more coming papers. CONCLUSIONS We have sought to extend the classical Drake equation to let it encompass Statistics and Probability. This approach appears to pave the way to future, more profound investigations intended not only to associate "enor bars" to each factor in the Drake equation, but especially to increase the number of factors themselves. In fact, thi s seems to be the only way to incorporate into the Drake UNCLASSIFIED/ ffOR OFFI&IAk Wlii QPII.¥ 53
UNCLASSIFIED/ /FO R OFFIEil.t.k W&i Q~lls¥ equation more and more new scientific information as soon as it becomes available. In the long run, the Statistical Drake equation might just become a huge computer code, growing up in size and especially in the depth of the scientific information it contained. It would thus be Humanity's fast "Encyclopaedia Galactica," Unfortunately, to extend the Drake equation to Statistics, it was necessary to use a mathematical apparatus that is more sophisticated than just the simple product of seven numbers. When this author had the honour and privilege to present his results at the SETI Institute on April lJlh, 2008, in front of an audjence also including Professor Frank Drake, he felt he had to add these words: " My apologies, Frank, for disrupting the beautiful simplicity of your equation," ACKNOWLEDGEMENTS The author is grateful to Drs. Jill Tarter, Paul Davies, Seth Shostak, Doug Vakoch, Tom Pierson, Carol Oliver, Paul Shuch and Kathryn Denning for attending his first presentation ever about these topics at the "Beyond" Center of the University of Arizona at Phoenix on February 8th , 2008. He also would like to thank Dan Werthi.mer and rus School of SETI young experts for keeping alive the interplay between experimental and theoretical SETI. But the greatest "thanks" goes of course to the Teacher to all of us: Professor Frank D. Drake, whose equation opened a new way of thinking about the past and the future of Human in the Galaxy. REFERE CES [I] http://en.wikipedia.org/wiki/Drake equation [2] http://en.wikipedia.org/wiki/SETI [3] http://en.wikipedia.org/wiki / Astrobiology [4] http://en.wikipedia.org/wiki/Frank Drake l5J Athanasios Papoulis and S. Unnikrishna Pillai, "Probability, Random Variables and Stochastic Processes", Fourth Edition, Tata McGraw-Hill, ew Delhi, 2002, ISBN 0-07-048658-l. [6) http://en.wikipedia.org/wiki/Gamma distribution [7] http://en.wikipedia.org/wiki/Central limit theore ill [8] http://en.wikipedia.org/wiki/Cumulants (9) http://en.wikipedia.org/wiki/Median [ I0]Jeffrey Bennett and Seth Shostak, "Life in the Universe", Second Edition, Pearson - Addison Wesley, San Francisco, 2007, ISBN 0-8053 - 4753-4. See in particular page 404. UNCLASSIFIED/ /FOR 8FFI&l.t.k Wii ,>DIis¥ 54
UNCLASSIFIED/ /FOR OFFIEil.t.k W&i Q~lls¥ References [1] Benford, Gregory, Jim and Dominic, "Cost Optimized Interstellar Beacons: SETI", arXiv.org web site (22 Oct. 2008). [2] Carl Sagan, "Cosmos", Random House, New York, 1983. See in particular the pages 298-302. [3] Bennet, Jeffrey, and Shostak, Seth, "Life in the Universe", second edition, Pearson - Addison Wesley, San Francisco, 2007. See in particular page 404. [4] C. Maccone, "The Statistical Drake Equation", paper #IAC-08-A4.1.4 presented on October 1st, 2008, at the 59 th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29 th thru October 3rd, 2008. UNCLASSIFIED/ /FOR OFFIEil.t.k W&i QNls¥ 55