DISCLOSURE / FILE
A 55-page Defense Intelligence Reference Document (11 March 2010) produced under the DIA's Advanced Aerospace Weapon System Applications program, walking through Frank Drake's 1961 equation and the author's statistical reformulation of it for estimating the number and distance of communicating civilizations in the galaxy.
A 55-page Defense Intelligence Reference Document (11 March 2010) produced under the DIA's Advanced Aerospace Weapon System Applications program, walking through Frank Drake's 1961 equation and the author's statistical reformulation of it for estimating the number and distance of communicating civilizations in the galaxy.
The DIA released this reference document on 11 March 2010 (information cutoff 1 December 2009) as one report in its FY 2009 Advanced Aerospace Weapon System Applications (AAWSA) series, the program tied to AATIP. It reviews SETI's history since 1959, reproduces Carl Sagan's unabridged 1983 description of the seven-factor Drake equation, and argues the classical equation is too simplistic because it returns a single integer for N. The author's fix, first presented as paper IAC-08-A4.1.4 at the 2008 International Astronautical Congress in Glasgow, replaces each of the seven variables with a random variable carrying a mean and a standard deviation, then solves the result using the Central Limit Theorem. The document notes that in the first 50 years of SETI searches, no extraterrestrial civilization was found.
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.p.4
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.p.4
Suppose next that N=1000, i.e. there are about a thousand extraterrestrial communicating civilizations in the whole galaxy right now.p.6
We define an advanced civilization as one capable of radio astronomy.p.8
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.p.10
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 (50 millions, in the above example).p.11
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UNCLASSIFIED / /#@R-OFFEGHH=TGEONE Defense Intelligence Reference Document Fo ee Acquisition Threat Support 11 March 2010 ICOD: 1 December 2009 An Introduction to the Statistical Drake Equation UNCLASSIFIED / (PO? OPPECEHEUSE OnE
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UNCLASSIFIED / /POROPFPLCIRE eStore
An Introduction to the Statistical Drake Equation
Prepared by:
(b)(3):10 USC 424
Defense Intelligence Agency
Author:
(b)(6)
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
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|Advanced Aerospace
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uestions pertaining to
this document should be addressed to |(b)(3):10 USC 424;(b)(6)
Manager, Defense Intelligence Agency, ATTN: |{b)(3):10 USC 424 [Bid
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UNCLASSIFIED / /#@R-OFFEGHH=TGEONE Defense Intelligence Reference Document Fo ee Acquisition Threat Support 11 March 2010 ICOD: 1 December 2009 An Introduction to the Statistical Drake Equation UNCLASSIFIED / (PO? OPPECEHEUSE OnE
UNCLASSIFIED / /POROPFPLCIRE eStore
An Introduction to the Statistical Drake Equation
Prepared by:
(b)(3):10 USC 424
Defense Intelligence Agency
Author:
(b)(6)
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, |(b)(3):10 USC 424
|Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments org
uestions pertaining to
this document should be addressed to |(b)(3):10 USC 424;(b)(6)
Manager, Defense Intelligence Agency, ATTN: |{b)(3):10 USC 424 [Bid
DC 20340-5100.
, AAWSA Program
ig 6600, Washington,
UNCLASSIFIED / / POT@GPRICRRAE USE OnEUNCLASSIFIED / (FP @fCOFFEGHirlGa-Obia= 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 1°, 2008, at the 59" International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29% thru October 3, 2008. 55 UNCLASSIFIED / (FP@?e @PFEGHEUSsE OnE
UNCLASSIFIED / / F@R-OFFECIRE USE ONCT Contents 1. INtroduction ....ccssccsssesceneneesenensnes serene caeananeeuzanenstananananasenauanezaraserinena senaueneuanananenen iv 2. The Key Question: How Far are They ? .....scsssercresecvcuvecsesencursensererers vuveveverecae wveree 4 3. Computing AN By Virtue of the Drake Equation (1961) .....ccccccsssccsnerenenenenneccsnenenes 7 4. The Drake Equation is Over-Simplified ......... a vuvuvavuuereceresccececevesecusercvasscurevezesurse 10 5. The Statistical Drake Equation ........ccceesssceceeeeeseee ence eeeeeeaeeenneneecedeaneesuenneeneeouneens 11 6. Solving the Statistical Drake Equation By Virtue of the Central Limit Theorem (CLT) Of Statistics ...ccscsescrcrenenenenancncnenenencnenanenananenanens sanananenesens pe soveecnanccenses penenee 13 7. An Example Explaining the Statistical Drake Equation ...... 8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical Drake Equation, ..escsescscsccesesccesescvevesesevevusevecasesavaveseonresesuscsusesuseseserevesesesusesesereses LO 9, The “Data Enrichment Principle” as the Best CLT Consequence Upon the Statistical Drake Equation (Any Number of Factors Allowed) .....:sccessssecesennnenees 23 10. CONCIUSIONS ....cccvesererccureecarerucecevavenavessscsessvusevavsraperevevaveneperaseraserepesuvercsurtrursres 2S Appendix A: Proof of Shannon’s 1948 Theorem Stating That the Uniform Distribution is the “Most Uncertain” One Over a Finite Range of Values wo eeecesseeseneeeneeeerecee Pere rte rey sueeeeeeeenaene fue ee eean ee ener nee = 25 Appendix B: Original Text of the Author’s Paper #IAC-08-A4.1.4 Entitled the Statistical Drake Equation ....scccssssseusssseessenereuanenesanecaneueuaveueuaesnanerenes 28 References ......... fo eeaueu ee sneeanauae snes ceseneseenenenenenenen a neavecseeseenene wenenene sensnenenucuarnerenessnenee 55 iii UNCLASSIFIED / /@R=-@FFECERESSEONTT
UNCLASSIFIED / /GQ2.Q55LGLAia SE Oia An Introduction to the Statistical Drake Equation i. 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 encugh! 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 | radictelescopes,” 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 the Solar System in the galaxy? This question was first faced in a scientific manner back in 1961 by the same scientist who also was the first experimental SETI radio astronomer ever: the American, Frank 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 galaxy measuring some 100,000 light years in diameter and some 16,000 light years in thickness of the Galactic 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, Reams 1s about 50,000 ly. UNCLASSIFIED / / b@.2:@5RIGESEUGH Oia
UNCLASSIFIED / /2@Re@bFEGIAi-UGE-Ohi
The thickness of the Galactic Disk at half-way from its center, Agi... iS about 16,000 ly.
The volume of the galaxy may then be approximated as the volume of the
corresponding cylinder, i.e.
2
Veta = 2 RGataw lt + (1)
Now consider the sphere around us having a radius r. The volume of such a sphere is
4_{ET_Distance \
Voour_Sphere = 3 —7L
(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 aff 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 shail take it for granted, although we are aware that this is a weak point in the
reasoning.
Furthermore, let us denote by NW the total number of civilizations now living in the
galaxy, including ourselves. Of course, this number NW is unknown. We only know that
N21 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:
Vecata yo Your Sphere (3)
N 1
That is, upon replacing both (1) and (2) into (3):
4 [EPitaneey
TReatay't _ 3 2
N 1 , F
(4)
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 / (P@R-OFFPGEH=USE-Oha=—UNCLASSIFIED / /F@feOFFIGHirUGE-Olinie ET_Distance() = Wun. © (5) where the positive constant C is defined by C= yo Révateay MGatay * 28845 light years « (6) Equations (5) and (6) are the starting point to understand the origin of the Drake equation that we discuss in detail in Section 3 of this 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 V may not be smaller than 1, i.¢., N21. Suppose then that we are alone in the galaxy, i.e., that V=1. Then the distance law (5) yields as distance to the nearest civilization from us just the constant C, i.e., 28,845 light years. This 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 pointing the antennas towards the Galactic Center, looking for beacons (see, for instance ref. [1]). * Suppose next that N=1000, 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 years. 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. e Suppose finally that V=1000000, i.e., there are a million communicating civilizations now in the galaxy. Then the distance law (5) yields an average distance of 288 light years. This is within the (upper) range of distances that our current radiotelescopes may reach for SETI searches, and that justifies all SETI searches that have been done so far in the first fifty years of SETI (1960-2016). In conclusion, interpolating the above three special cases of NV, we may say that the distance law (5) yields the following key diagram of the average ET distance vs. the assumed number of communicating civilizations, N, in the galaxy right now (Figure 1): UNCLASSIFIED / /-P@R:OEELGHAHUGEOhia==
UNCLASSIFIED / / F@R-O@FFEGEHEUSEONtTT™” Average DISTANCEof the nearest ET civilization vs. the ASSUMED NUMBER of ET civilizations in the Gak 200¢ 1750 1501 1006 750) N S00} a 25 Average DISTANCE of the civilizaéon nearest to us in LIGHT YEARS 0 0 toooee 200000) 300000) 400000 500000) 600000) 700000 800000 900000 1000000. ASSUM ED NUMBER of civilizations in the Galaxy (that is, N in the Drake equation) Figure 1. DISTANCE LAW; i.e., the Average Distance (plot along the vertical 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 Donald Drake (born 1930) to estimate the numerical value of WV. 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 cosmic 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 night, near one of those faint pinpoints of light is a world on which someone quite different from us is then glancing idly at a star we call the Sun and entertaining, for just a moment, an outrageous speculation. UNCLASSIFIED / / R@2=@ EIGER GEO hee
UNCLASSIFIED / /208-@5EIGEAL-UGEON RS 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. This is, of course, a parochial if essential definition. There may be countless worlds on which 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: e Ns, the number of stars in the Milky Way galaxy. e fp, the fraction of stars that have planetary systems. e ne, the number of planets in a given system that are ecologically suitable for life. e ff, the fraction of otherwise suitable planets on which life actually arises. e ff, 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 technical civilization develops. e fL, the fraction of planetary lifetime graced by a technical civilization. Written out, the equation reads N=Nos- fo-ne- fl fis fo fh (7) All of the fs are fractions, having values between 0 and 1; they will pare down the large value of Ns. To derive NV we must estimate each of these quantities. We know a fair 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. 8 UNCLASSIFIED / /POROPPICERE USE One
UNCLASSIFIED / SPOR OPPTrCTRE USE -Oner 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 101. 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 equal to 1/3. Then the total number of planetary systems in the galaxy would be Ns fp ~ 1.3 x 101 (the symbol ~ means “approximately equal to”). If each system were to have about ten planets, as ours does, the total 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 that may be suitable 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 suitable for life becomes Ns fp ne ~ 3 x 107. Experiments show that under the most common cosmic conditions the molecular basis of life is readily 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 ff ~ 1/3, implying a total number of planets in the Milky Way on which life has arisen at least once as Ns fp ne ff~ 1x 104, a hundred billion inhabited worlds. That in itself is a remarkable conclusion. But we are not yet finished. The choices of ff and fc are more difficult. On the one hand, many individually unlikely steps had to occur in biological 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 capabilities. Considering the apparent difficulty in the evolution of large organisms, represented by the Cambrian explosion, let us choose ff x fc = 1/100, meaning that only 1 per cent of planets on which 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 think that, even given ten or fifteen billion years, the evolution of a technical civilization is unlikely. This is not a subject on which we can do much experimentation as long as our investigations are limited to a single planet. Multiplying ' Carl Sagan was writings these lines back in the 1970’s, when no extrasolar planets had been discovered yet. The first such discovery occurred in 1995, when 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 $1 Peg B. Many more extrasolar planets were discovered around nearby stars ever since. As of April 2009, 347 extrasolar planets (exoplanets) are listed in the Extrasolar Planets Encyclopaedia. UNCLASSIFIED / -2@R-@ERLGE Als SOD ves
UNCLASSIFIED / /G@2-855GLAlrUGEONE= these factors together, we find Ns fp ne ff fi fc ~ 1 x 10°, 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 civilization? 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 1/108, 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 Ws fp ne fi 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 WV 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, NV ~ 107, and the number of extant civilizations in the galaxy is in the millions. Thus, for all our concern about the possible unreliability 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 realistic 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 NV, 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 10 UNCLASSIFIED / /#@irOPEGHrUGhGhint
UNCLASSIFIED / / P@fe@FEEGitir-UGE-OlinG= million or even billions in the optimist’s opinion. A lot of uncertainty is thus affecting our knowledge of NV as of 2010. In all cases, however, the final result about V 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 this author regarded as “too simplistic” and improved mathematically in his paper #IAC-08-A4.1.4, entitled “The Statistical Drake Equation” and presented on October 1**, 2008, at the 59%" International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29" thru October 3™, 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 explain 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 example. Consider the first independent variable in the Drake equation (7), i.e., Vs, 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) a// 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 Vs = 350 x 106, 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 (50 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 statistical tool for the investigation of a host of facts hardly known to us in detail. In other words still: e 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). e But now we also ADD A STANDARD DEVIATION o,, on each side of the mean value, that is provided by the knowledge gathered by scientists in each discipline encompassed by each D,. 11 UNCLASSIFIED / /F@R-@FFEGEREUSE-ONEY
UNCLASSIFIED /p&@E-@EEEGE nS E-Cley= Having so done, the next question is: How can we find out the PROBABILITY DISTRIBUTION for each p,? 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 this apparent conundrum comes from Shannon’s Information Theory and a thearem 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 », 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 y, and of the standard deviation c,, automatically determines the RANGE of that random variable in between its lower (called a;} and upper (called 4, ) limits: in fact these limits are given by the equations ( = Mp, -V3ap, (8) b= My, + Boy (the “surprising” factor v3 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 yp, 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 »,, what is the probability distribution of NV , given that NV is the product (7) of all the p, ? In other words, not only do we want to find the analytical expression of the probability density function of NV, but we also want to relate its mean value y, to all mean values Hp, Of the D,, and its standard deviation o, to all standard deviations «, of the D,. 12 UNCLASSIFIED / /®@fe@FE5GRAirnUGE=Q ahem
UNCLASSIFIED / (-@R-OFFEGHirUGE-O Ne 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 variables p, . 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 4 in the statistical Drake equation was found by this author in September 2007. The key steps are the following: e Take the natural logs of both sides of the statistical Drake equation (7). This changes the product into a sum. e 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. e Thus, the natural log of N tends to a Gaussian. e Thus, WV tends to the LOGNORMAL DISTRIBUTION. e The mean value and standard deviations of this lognormal distribution of W may all be expressed analytically in terrms of the mean values and standard deviations of the logs of the bp, 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 1%, 2008. This original paper is reproduced in Appendix B. To sum up, not only is it found that V 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 condition that the number of terms in the product (7) must be FINITE. 13 UNCLASSIFIED / (P@feOFEEGHIIHUGE-ONiiee=
UNCLASSIFIED / /F@R-GEEIGEALAUGE OMe 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 NV 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. 14 UNCLASSIFIED / 4§@P:@EEIGERE USE ONE
UNCLASSIFIED / / B@R:@EGhrnUGE-Oiineam 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 Probability distribution Lognormal Probability density function 1 4 (inlu) a fx(n)=—- e ** (20) n 20 Mean value o (N)=e"e 2 Variance 3 4 z 2 oy =e" e* Qe -1) Standard deviation a Gy =e" e? ve™ -1 All the moments, i.e. k-th moment ne (n*)= ee 2 Made (= abscissa of the lognormal peak) = — pt - Mrante = Mpeak = & Value of the Mode Peak 5 = “Hg? . . 1 Foy Pate) = To’ Median (= fifty-fifty probability value for median = 11 =e“ N) Skewness Ky | o 2) eye vet > Nf ae To ay (Ky): be" -1) ( +3e°7 +607 +6) Kurtosis Ka a=? 42 432" 6 (Ki) Expression of win terms of the lower (a) 7 1» [in(e,)—1]—«; [in(,)-1] and upper (01) limits of the Drake w= D)= uniform input random variables D; b-4; Expression of cin terms of the lower (a/) and upper (81) limits of the Drake uniform input random variables Dj oz Ye} = yy _ Gib nl, }- nla, I (; — a y 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: e The first column on the left lists the seven input sheer numbers that also become * The mean values (middle column). e Finally the last column on the right lists the seven input standard deviations. 15 UNCLASSIFIED / (P@feO@FPEGHirUGE Ohne
UNCLASSIFIED / /#@@FEIG@hH=US=ONEY 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. Ns := 350-10" UuNs c= Ns GNs := 1-10" . _ £0 a . 10 = ‘= oip = — P= 356 Mp = fp 100 1 ne ‘= 1] [ine ‘= ne Gne = —= a ui =a oft = 22. 100 100 g-= 2 esa _ 10 ~ 100 was ~ 100 5) a 9 fe = — ufc 7= fe cfc = — 100 00 _ 10009 wea .- 1009 10°" 10 Ne Nofp-ne-fteaL X = 3500 Table 2. Input Values (i.e. mean values and standard deviations) for the Seven Drake Uniform Random Variables Di . The first colurnn 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 WV = 3500. In fact, a MathCad code written by this author and capable of performing all the numerical calculations required by the statistical Drake equation for a given set of seven input mean values plus seven input standard deviations, yields for V 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 Range = Mock = OY e* = 250 (this is equation (99) of Appendix B), while the median (fifty- fifty value splitting the lognormal density in two parts with equal undergoing areas) falls at ADOUt Arian =“ = 1740 . These seem to be smaller values than WV = 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 W is the MEAN VALUE! Now if you look at the thin curve in Figure 2 below (i.e. the lognormal distribution arising 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 16 UNCLASSIFIED / /POROPPICIRE SSE ORE
UNCLASSIFIED / /#@R@FEEGitirUGE-O2iaiamm surprised to find out that it equals (V)=e"e¢? ~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 extraterrestrial civilization! 4 PROBABILITY DENSITY FUNCTION OF N ot LAS 3-107 N i to — Prob, density function of N ww S A 0 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 Resarting 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 oy =e” ¢ ? ve" -1=11195 and so the expected number of WV 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. 17 ‘UNCLASSIFIED / /®@-OFFEGEE USEF ONE
UNCLASSIFIED / / POT OPrF2e?=-vsr ent 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 V 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 . 3 I . Fet_distane 1) = Gino -¢@ 2 0) holding for r20. This is equation (114) of Appendix B. Starting from this equation, the MEAN VALUE OF THE random variable ET_DISTANCE is computed as Le (ET_Distance) = Ce Fel (10) which is equation (119) of Appendix B, and finally the ET_DISTANCE STANDARD DEVIATION Wer pistme =Ce Fel’ Ve? -1 (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 result 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: ues r, =Ce 4¢!8 + 2,670 light years (12) mean value 18 UNCLASSIFIED / /PO@R°OFRGErUGrOhirte=
UNCLASSIFIED / APOTeOPPICIAL USE ONTT which is equation (153) of Appendix B, and ui os o Orr Disune =Ce 2 ¢'* Ve? —1 ~ 1,309 light years (13) which is equation (154) of Appendix B. 19 UNCLASSIFIED / /-@2-QERGiArDGEONire=
UNCLASSIFIED / (POROOPPTPCH=UGEONe 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 galaxy volume. Probability distribution Unnamed Probability density function : 3 0] ! Fet_pistane (1) = =* == ¢ *e r V2n0 N ical rel he Milk’ ste Rp? ‘ . umerical constant C related to the MHIKY | coe Rj Ray © 28845 Tight year Mean value noo (ET_Distance) = Ce 3 gi8 Variance 2 o s 2 arias ool ou — SET viswne =Co@ * e? fe? 1 Standard deviation ae Bf ot =Ce3 Ye 9 _ rr pisune =Ce 7! Ve? 1 All the moments, i.e. k-th moment Hoe . gy (ET_Distance* ) =Che Fe Mode (= abscissa of the lognormal peak) 2 ee = = 3g Y Trade = Pye =Ce 7 Value of the Mode Peak Peak Value Of fir pistane(*) = 3 via oe = fer pisune Unde) = =e ee" CV2no Median (= fifty-fifty probability value for N) e median =m=Ce 3 Skewness eo ket e“let -3e' 120% K3 7 3 (Ky h Bar se 4c? a 20° \2 Oye % -4e ° -3¢ % 41263 -6e ? Kurtosis K 4a eS 2a" tose 9 +2¢343¢9 -6 (K2) Expression of “in terms of the lower (ai) _< v)< 2b [in(o,)- 1]-a,[in(a, )—1] and upper (bi) limits of the Drake uniform u= dX ‘= eee input random variables Di b,-a; Expression of 2” in terms of the lower (ai) and upper (bi) limits of the Drake uniform input random variables Di 2 -Y'03 = 5 1-elnG)- ole it i-l (, —4G y 20 UNCLASSIFIED / / POT OPPICRRE USE OnE
21 UNCLASSIFIED / / F@?e@FPEGie SE ONEY UNCLASSIFIED / / P@R-@FFEGHirUGE-Ohia
UNCLASSIFIED / EO. @EEIGLAL-UGE-O hie It is clarifying to draw the graph of the ET_Distance probability density (9): 563-107 DISTANCE OF NEAREST ET_CIVILIZATION 45-10" 20) had bo io) 2 2) 2.2510 Probability density function (1 /meters) 113-10? =“ 0 500 1000-1500 = 2000» 2500 3000 «3500 «4000 =4500 = 5000 ET_Distance from Earth (light y cars) 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 fer pisame(?) 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 Hie 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 ae Foude = Fyre =C@ Fe % =1,933 light yours. (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 value of the probability distribution (9) for fer pictane(7). 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 eo Mean value iS ten cue = Ce 2 e!* = 2670 light years. This is the MEAN (value of the) DISTANCE at which we can expect to find extraterrestrials. 22 UNCLASSIFIED / fF@R-OFFRECEHE=USE-OhE>
UNCLASSIFIED / AF@TeOFFECEHHEUSE-ONEY
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 extraterrestrials 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),
a SF a
Gir pitme =C€ 2! Ve? —1 = 1309 light years.
More precisely, this is the so-called 1-sigma (distance) level. Probability theary then
shows that the nearest extraterrestrial 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 fry picune("
taken in between these two lower and upper limits, that is:
Fir nistune {r)dr = 0.75 = 75% (15)
ewe
1
36 lightyears
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.”
23
UNCLASSIFIED / (POTCOFFEGEAHUGE-OieeUNCLASSIFIED / /(PO@?PR-OPPPCREUSEOnrr 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. 24 UNCLASSIFIED / /#@fe@FPEGHirUGE-O Ni
UNCLASSIFIED / / F@R@FFEGiirUGE-ORiae=
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:
2 om Tee Bes:
S85 Kay.
A Mathematical Theory of Communication
ByC. 5, SHANNON
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):
4. Let pix be a one-dimentional distrrbution. The Form ef px
condition that the standard devianen of x be fined ata it
giviag a maximum entropy zulyect to the
auisian. To show this we must maximize
Aix: - { pixitogpiidy
with . .
wa / pourdy and ot - | Boxidy
as coastaints, This requres. by the calculus of variations, maxtmuzing
/\ piviloggix, b Ap + piv! ax.
The conditon for this is
3
L legvar- ur - a — 0
and consequently (adyusting the constants to satisfy the constraints}
and
25
UNCLASSIFIED / / P@R-@FFECERE "USE ONEYUNCLASSIFIED / /B@Re@QEREGIAieUGE=O Misano °. Tfxis hanted to a bal? ine (piar — 6 for x +! Oy and the first moment of y is fixed at a a7 i pix dy. #37 then the masumum enopy occurs when and 1s equal to log 22. 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 probability density function p(x) is limited to a FINITE INTERVAL a<.x <6. This is obviously the case with any physical POSITIVE random variable, such as a distance, or the number AW 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<x<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 statistical Drake equation, it sounds like: “Since we don’t know what the probability distribution of any one of the Drake random variables », is, it is safer to assume that each of them has the maximum possible entropy overa, <x<6,, i.e., that D, is UNIFORMLY distributed there. The proof of this theorem is along the same lines as for the previous two cases discussed by Shannon: We start by assuming that a, <x<#,. We then form the linear combination of the entropy integral plus the normalization condition for D, BE ple)tog nlx)+2 plalJax=0 where 2 is a Lagrange multiplier. Performing the variation, one finds —log p(x)-14.4 =0 that is: p(x)=e7}. Applying the normalization condition (constraint) to the last expression for p(x} yields by b, b, [= f P(x) ax =f 7 de aa dx =e? "(b; -a;) a ; A fy that yields 26 UNCLASSIFIED / /SQRQEELGE Abed IG Fil bbe
UNCLASSIFIED / (POTeOPPERCERE USE OnE” and finally px)= ! with a, <xsb, b, -a i showing that the maximum-entropy probability distribution over any FINITE interval a, <x<b, is the UNIFORM distribution. 27 UNCLASSIFIED / ,5EO2. O5ELGiireSEONET
UNCLASSIFIED / / b@2-O5EEGHN=U6GEONEY Appendix B: Original Text of the Author’s Paper #IAC-08- A4.1.4 Titled the Statistical Drake Equation TAC-08-A4.1.4 THE STATISTICAL DRAKE EQUATION Claudio Maccone Co-Vice Chair, SETI Permanent Study Group, International Academy of Astronautics Address: Via Martorelli, 43 - Torino (Turin) 10155 - Italy URL: hup://www.maccone.com/ - E-mail: clmaccon@libero.it ABSTRACT. We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation 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. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form ol the CLT, depending on (he mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: 1) 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 N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal V 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 for each factor. This is both physically realistic and practically very useful, of course. 3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizalions 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 variables in the Statistical Drake Equation is compatible with the CLT. Sv, 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 inslance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 350 billions with a standard deviation of (say) 1 billion. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows 28 UNCLASSIFIED/ (POT OP PICIAE USE ONT
UNCLASSIFIED / (FO@FeOFFECHi=UGE-OMinG==e
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 ExtraTerrestial
Intelligence, see ref. [2]) and Astrobiology (see ref.
[3]). Devised in 1960, 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= No: fpones fl fie for fl. QQ)
Where:
1) 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) flis fraction ( percentage) of such “ready for
life” planets on which life actually starts and
grows up (but not yet to the “intelligence”
level).
5) fi is the fraction (= percentage) of such
“planets with life forms” that actually evolve
until some form of “intelligent civilization”
emerges (like the first, historic human
civilizations on Earth).
6) fe 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 radioastranomy
{at least).
7) fL is the fraction of galactic civilizations alive
at the ime when we, poor humans, attempt to
pick up their radio signals (that they throw out
into space just as we have done since 1900,
when Marconi started the transatlantic
transmissions), In other words, fL is the
29
number of civilizations now transmitting and
recciving, and this implics 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 civilization? Or are they
slowly becoming wiscr, 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. Bul 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 Dj.
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 lo this paper,
please.
2. STEP 1: LETTING EACH FACTOR
BECOME A RANDOM VARIABLE
In this paper we adopt the nolations of the
great book “Probability, Random Variables and
Stochastic Processes” by Athanasios Papoulis
(1921-2002), now re-published as Papoulis-Pillai,
UNCLASSIFIED / /POReOPFICTRE USE OUNCTUNCLASSIFIED /PF@fOFEEGii=UGE-O Nm
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 “deterministic”) 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 Institute (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
D,=Ns
Dy = fp
Dy=ne
D,= fl (2)
Ds = fi
Dy = fe
Dy = fh
so that our STATISTICAL Drake equation may be
simply rewrilien as
30
v=]. 3)
fal
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 (posilive) mean value and standard deviation...
... the natural question then arises: how are the seven
mean values on the right related 10 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 yariables are nol casy to
handle in probability theory. It is actually much
casier 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 probability
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 / /EQR.DEEEGI idl dG a@ pieiime31
UNCLASSIFIED / (FOR OPPTCTRE USE One
| 2. Step 1: Letting each factor become a random |
| 2.1. Step 2: Introducing logs to change the product into a
| 2.2, Step 3; The transformation law of random variables. |
3, Step 4: Assuming the easiest input distribution for
each 9: the uniform distribution.
variables D;.
3.1. Step 5: A numerical example of the Statistical Drake equation
with uniform distributions for the Drake random
f
3.2. Step 6: Computing the logs of the
7 uniformly distributed
Drake random variables
Bi.
it
3.3, Step 7: Finding the probability
density function of 4, but
only numerically not
analytically.
DEAD END!
4. The Central Limit Theorem (CLT) of Statistics,
S. LOGNORMAL distribution as the probability
distribution of the number # of
communicating ExtraTerrestrial Civilizations
in the Galaxy.
6. Comparing the CLT results with the Non-CLT
results, and discarding the Non-CLT approach.
7. DISTANCE to the nearest ExtraTerrestrial
Civilization as a probability distribution (Paul
Davies dubbed that the Maccone distribution).
7.4 Classical, non-probabilistic derivation of the
Distance to the nearest ET Civilization,
I
7.2 Probabilistic derivation of probabillty density
function for nearest ET Civilization Distance.
7.3 Statistical properties of the distribution.
7.4 Numerical example of the distribution.
{
8. DATA ENRICHMENT PRINCIPLE as the best
CLT consequence upon the Drake equation:
any number of factors aflowed for.
UNCLASSIFIED / SPOT OTFICTRE USE ONCUNCLASSIFIED / /POR'GPPICEE SSE Oh
So, let us take the natural logs of both sides of the
Statistical Drake equation (3) and change il into a
sum:
mv) =f [] >| = Soto. (4)
ft fal
It is now convenient to introduce eight new (positive)
random variables defined as follows:
Y =In(V
[y= 6
ly, =In{D,) i=1....,7.
Upon inversion, the first equation of (5) yields the
important equation, that will be used in the scque!
N=e’. (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 distribution shall we attach to
each of the seven (positive) random variables D,'?”
It is not casy to answer this question because we
do not have the least scientific clue to what
probability distributions fit at best to cach 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
cuo)= |
fx (ay. ) Jono
é
has its independent variable y ranging between oc
and 9: and so it can apply to a real random variable
¥ only, and never to positive random variables like
those in the statistical Drake equation (3). Period.
Searching again for probability density functions
chat 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
32
type of probability density function (pdf) for the last
seven of cquations (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 fimction Y= g(X) of another random
variable X having a known pdf. In other words, if the
pdf fy (x) of a certain random variable X is known,
then the pdf f,(y) of the new random variable Y,
related to X by the functional relationship
¥ = (x) (8)
can be calculated according to this rule:
1) First invert the corresponding non-probabilistic
equation y= gx) and denote by x, G) the
various real roots resulting from the this
inversion.
2) Second, take notice whether these real roots may
be either tinitely- or infinitely-many, according
to the nature of the function y = g(x).
3) Third, the probability density function of ¥ is
then given by the (finite or infinite) sum
fx (oi(v)
LOD 9)
i
where the summation extends to all roots x; (y) and
\s'(x(v)] is the absolute value of the first
derivative of g(x) where the i-th root x,(¥) has
been replaced instead of x.
Since we must use this transformation law to transfer
from the D, to the Y,; =In(D,), it is clear that we
need to start from a 2), 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. In conclusion, we
discarded the gamma distributions and confined
UNCLASSIFIED / AEQR.OEELCT OL old SEN MeesUNCLASSIFIED / #P#@?e@EFEGiirU GEO bine
ourselves to the simpler uniform distribution instead,
as shown in the nest section.
4. STEP 4: ASSUMING THE EASIEST
INPUT DISTRIBUTION FOR EACH 2;:
THE UNIFORM DISTRIBUTION
Let us now suppose that each of the seven Dj; is
distributed UNIFORMLY in the interval ranging
from the lower limit a,>0 to the upper limit
b, 2 4;.
This is the same as saying that the probability
density function of each of the seven Drake random
variables D, has the equation
anion p, 2) = ; with 0S; $x<b, (10)
=a,
i i
as it follows at once from the normalization condition
te
[ Sunitrmb, (x) dx=1. a 1)
Let us now consider the mean value of such
uniform D; defined by
fy,
! ) xdx
yo
{uniform_D;) = i} * furitemy, (a) ax = ba, Jer
ty a a
, 2 2
1 B be -ap az th;
~ b-a;| 2 2{b,-a;) 2
By words (as it is intuitively obvious): the mean
value of the uniform distribution simply is the mean
of the Jower plus upper limit of the variable range
(uniform_D,) = aoe | (12)
In order to find the variance of the uniform
distribution, we first need finding the second moment
. 2 fy
{uniform D 7 ) = ) e Fanitom_D, («)ex
1 fh 3 1 felt bs -a}
= i) av dx= —| =-—-—,
b, a; a, ba; 3] 3(b:-a;)
é i i
33
(b, -a;) (2 +a,b, +b) ) a} +a,b, +?
= - £ = .
3(b, -a;) 3
The second moment of the uniform distribution is
thus
2 Lajb, +bF
(uniform_D;?) = S282 a L (13)
From (12 and (13) we may now derive the variance
of the uniform distribution
FinitvenD, = (uniform_D;? ) —(uniform_D; y
4
_ a tab +b; (a th) _ (a)
14)
3 4 12 ‘
Upon taking the square root of both sides of (14), we
finally obtain the standard deviation of the uniform
distribution:
_ bj 4;
Franti, = a3 | (15)
We now wish to perform a calculation that is
mathematically trivial, but rather uncxpected from
the intuitive point of view, and very important for our
applications to the statistical Drake cquation. Just
consider the two simullancous cqualions (12) and
(15)
{aniform_D,) = fi = t
ba 6
Faniorno, = aa .
Upon inverting this trivial linear system, one finds
(a, = {uniform_D;)— V3 unitorm D, (17?)
1 = {uniform_D;) +3 Fantom, ‘
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 certain Drake random
variable Dj, then the lower and upper limits of the
relevant uniform distribution are given by the two
equations (17), respectively.
UNCLASSIFIED / /5@2-@5EIGEAaYGE-@hieeo=UNCLASSIFIED / / P@?e@FEEGhrb-UGE-OAi am
In other words, there is a factor of V3 = 1.732
included in the two equations (17) that is not obvious
at all to human intuition, and must indced be taken
into account.
The application of this result to the Statistical Drake
cquation is discussed in the next section.
3.1 STEP 5: A NUMERICAL EXAMPLE
OF THE STATISTICAL DRAKE
EQUATION WITH UNIFORM
DISTRIBUTIONS 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), In other words, we now
assume that one has:
(18)
9
Fonitormn, = 1°10".
{nfo )=350-10°
Therefore, according to equations (17) the lower and
upper limit of our uniform distribution for the
random variable Ns=D, are, respectively
yy = (uniform _D,)— 13 Ou pigomip, = 348-3-10” 9)
by, =(uniform_D,) + v3.0, = 351.7-10°
unilonn_D,
Similarly we proceed for all the other six random
variables in the Statistical Drake cquation (3).
For instance, we assume that the fraction of stars
that have planets is 50%, ic. 50/100, and this will be
the mean valuc of the random variable fp=D2. We
also assume that the relevant standard deviation will
be 10%. i.e. that o,, =10/100 . Therefore, the
34
relevant lower and upper limits for the uniform
distribution of fp=D2 turn oul to be
@ jy = (uniform_D) - V3 Oyniirm_p, = 0-327
i (20)
b yy = (uniform _D2)+ V3 Oynitem_p, = 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 1, 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/V3 =0.577 to compensate the 3
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
chat
yy = (uniform_D;)~V3 Oysinea.D, =0
(21)
bye = (uniform_D3) + V3 FunitomD, = 2
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 1.
Please notice that, had we assumed all the
standard deviations to cqual zera in Table 1, 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 oul lo be 3500:
[W=3500] ony
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 / /EQR.OEELCIAiNGEOhi>UNCLASSIFIED / /PO@?eOPPECEREH USE Ontr Ns i= 30-10" UNs = Ns GNs <= 1-10 —— hip := tp ofp = 22 = T05 ~ ~ 100 1 ne c=] Mne c= ne one c= 8 ° ace 2 jul c= 1 oft := 22 100 100 > aw 22 (ui = fi of = 22 100 100 - 20 - ~ - 10 te = —— Mfc := fe oie = — 100 100 fLi= a jal <= tL = a 10 10 N c= Ns-fp-ne-f-fi-fe L N = 3400 Table 1. Input valucs (i.c. mean values and standard deviations) for the seven Drake uniform random variables D;. The first column on the left lists the seven input shcer numbers that also become the mean valucs (middle column). Finally the last column on the right lists the seven input standard deviations. The bottom line is the classical Drake equation (1). 3.2 STEP 6: COMPUTING THE LOGS OF THE 7 UNIFORMY DISTRIBUTED DRAKE RANDOM VARIABLES D; Intuitively speaking, the natural log of a uniformly distributed random variable may aot be another uniformly distributed random variable! This is obvious from the trivial diagram of y =In(x) shown below: Natural logarithm of x z 2 4B i] 3 on e 0 z 3 g +t 3 i <4 Zz oO 1 2 3 4 5 POSITIVE independent variable x Figure 1. The simple function y = In(x). 35 So. if we have a uniformly distributed random variable D; with lower limit @,and upper limit 5;, the random variable (23) must have its range limited im between the lower limit fn(a;) and the upper limit /n(b;). In other words, this are the lower and upper limits of the relevant probability density function fy (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= a(x)= In) (24) That, upon inversion, yields the single root a(y)=ax(y)=e". (25) On the other hand, differentiating (24) one gets UNCLASSIFIED / £602-055tGEHrUOE-ONEY
UNCLASSIFIED / /#@®@FEIGhr=-UGE-ONE =
e()=2 and g(x (x= 5=+ 26)
xily) @
where (25) was already used in the last step. By
virtue of the uniform probability density function
(10) and of (26), the general transformation law (9)
finally yields
Fa (xy) Loe
OE uO] b, -a; —— . (27)
In other words, the requested pdf of ¥; is
In(a; )< ys Ino; ) (28)
Probability density functions of the natural logs of
all the uniformly distributed Drake random
variables Di .
v7. re
KOT is
This is indeed a positive function of y over the
interval Inta,)< ¥E In(t,), as for every pdf, and it is
casy lo sce that its normalization condition is
fulfilled:
Ine } Info} gt ltl) _ gina)
dy= dy = —__—_—_—_- =
Sic fe xb ) . ~ Ira —a; b; -4@;
(29)
Next we want to find the mean value and
standard deviation of Y, , since these play a crucial
role for fulure developments. The mean value iy, i) is
given by
_ In(b, ) Inf.) yee”
= [ny fiOo)ds v= 4
ne, }D, — a;
_4 [in@,)- t)-«fin(a;)—1) 80)
b, -4;
t t
This is thus the mean value of the natural log of ail
the uniformly distributed Drake random variables
Di
b,[in(, )-1]-a,fla(e, }— 1]
b; =;
{¥;) = (In (2; ) =
| G1)
36
In order to tind the variance also. we must first
compute the mean value of the square of Y,, that is
204)
fy? _ Lah, , 2 Tint yy “@ ty
Vi ) Baas? rad (r)dy = Jac 4; “
b, (in? )- 2; )+ 2|-a, [m2 (a, )- 2tn(@; + 2|
b; ,
i
~<a;
.-(32)
The variance of Y; = In(Di) is now given by (32)
minus the square of (31), that. after a few reductions,
yield:
a,b, [in{o, )- Ina; yr (33)
;-4,)°
Whence the corresponding standard deviation
2 2? _
Oy, = Oinin, = 1
b,[m(6, )-
Fy, = ayy =
Let us now turn to another topic: the use of
Fouricr transforms, that, im probability theory, are
called “character’ functions,” Following again the
notations of Papoulis (ref. [5]) we call “characteristic
function”, @)(¢) , of an assigned probubility
distribution Y; , the Fourier transform of the relevant
probability density function, that is (with j= ¥—-1)
oy G)= f° eA oar} 35)
The use of characteristic functions simplifics things
greatly. Por 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 resull 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 ¥; , i.e.
@y (C= f ef (va
In). » ee
v= [oe ) b-a
i i
UNCLASSIFIED / /5Q0:2.055i6hibold SEQ highsUNCLASSIFIED / /S@R-OFREGHU6EONEE
1 (ere (icy 1
= dyv=
| (a ichy pe )
b 74 ina;) b - 4; ‘THit
inte, }
ellhis lin) _ eli Nala, at Ho fa (6
= " 36)
Gari) © “ate
Thus, the characteristic function of the natural log
of the Drake uniform random variable Dj is given by
_ lage
~—a; dS
G7)
3.3 STEP 7: FINDING THE
PROBABILITY DENSITY
FUNCTION OF N, BUT ONLY
NUMERICALLY NOT
ANALYTICALLY
Having found the characteristic functions
o, (¢ ) of the logs of the seven input randont
variables D; . we can now immediately find the
characteristic function of the random variable Y =
In(A) defined by (5). In 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 by () cquals the product
of the seven «by (¢):
bie — alte
o,(0=[]ou(0 “gaits ano (38)
The next step is to dtvert this Fourier transform in
order to gel the probability density function of the
random variable Y = In(N). In other words, we must
compute the following inverse Fourier transform
_ bis
pl i
Ele le by; - a4) Thode | ae. G9)
37
This author regrets that he was unable to compute the
last integral analytically. He had to compute it
numerically tor the particular values of the 14 a; and
b; that follow from Table | and equations 17. The
result was the probability density function for Y =
In(N) plotted in the following Figure 2.
=, PROB. DENSITY FUNCTION OF Y=tn(N)
3.
3
2 03
= y
2 92 fA
g V4
2 ol 7
Z y |
z OMT 2 345 67 8 9 1001 12
Independent variable Y = In(N)
Figure 2. Probability density function of Y = In(N)
computed numerically 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 = In(N), or
alternatively, that N=cxp(Y), as stated by (6). We
must thus resort to the transformation law of random
variables (9) by sctting
yee(xdse*. (40)
This, upon inversion, yields the single roat
ai(y)=(¥)=In(y). (an
On the other hand, differentiating (40) one gets
e(ee and ¢ ‘(x(y)) = eluly x) =) (42)
where (41) was already used in the last step. The
general transformation law (9) finally yields
(xy
fuls)= ea AT fr Qn{y). 43)
UNCLASSIFIED / / F@R=@FFEGEUSEONEYUNCLASSIFIED / (POR OPFPLCEE= US E-ONE
This probability density function f,(¥) was
computed numerically by using (43) and the numeric
curve given by (39), and the result is shown in Figure
{PROBABILITY DENSITY FUNCTION OE N
4-10
AN
~
Pe,
7.
2
5,
arn
0 1000-2000 =—3000 4000
N= Number of ET Civilizations in Galaxy
Prob. density function of N
0
Figure 3. The numeric (and not analytic) probability
density function curve fy(y) of the number N of
ExtraTerrestrial Civilizations in the Galaxy according
to the Statistical Drake equation (3). We sce that the
curve peak (i.c. the mode) is very close to low valucs
of N, but the tail on the right is high, meaning that the
resulting mean value (nN) is of the order of
thousands.
We now want to compute the mean value (nN )
of the probability density (43). Clearly, it is given by
()= f y fyly)dy. (44)
a
This integral too was computed numerically, and the
result was a perfeet match with N=3500 of (22), that
is
(W) = 3499,99880 177509 + 0.00000012 49146861 (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
Oy = 3953.42910 143389 + 0.00000003 2800058: . (46)
38
This standard deviation, higher than the mean value,
implics 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 |.
We conclude that, unfortunately, even under the
simplifying assumptions that the Di be uniformly
distributed, it is impossible to solve the full problem
analytically, since all caleulations 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 fy(y) ?
Rather surprisingly, the answer is “no”, and there
is indeed a way out of this dead-end, as we shall sec
in the next section.
5. THE CENTRAL LIMIT THEOREM (CLT)
OF STATISTICS
Indeed there is a good. approximating analytical
expression for fy(v), and this is the following
lognormal probability density function
_linls at
e 2 cy2zo)/ (47)
ano
1
ty (rsa o)= > .
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
1901 by the Russian mathematician Alexandr
Lyapunov (1857-1918), and later (1920) by the
Finnish mathematician Jarl Waldemar Lindeberg
(1876-1932) under weaker conditions. These
conditions are certainly fulfilled in the context of
the Drake cquation 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 referred for more details, such as the equations
for the Lyapunev and the Lindeberg conditions,
making the theorem “rigorously” valid.
UNCLASSIFIED / A2@R-@ERELGHr=UEE OEYUNCLASSIFIED / / F@fe@FPiGEi=UGrOhin Put in loose terms, the CLT states that, if one has a sum of random variables even NOT identically distributed, this sum tends to a normal distribution when the number of terms making up the sum tends to infinity. Also, the normal distribution mean value is the sum uf 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 can possibly be. In other words, the CLT applied to the Statistical Drake equation (3) leads immediately to the following three equations: 1) The sum of the (arbitrarily distributed) independent random variables ¥, makes up the new random variable Y. 2) The sum of their mean values makes up the new mean value of ¥, 3) The sum of their variances makes up the new variance of ¥, In equations: v=) a8 of - Sot fl This completes our synthetic description of the CLT for sums of random variables. 6. THE LOGNORMAL DISTRIBTION IS THE DISTRIBUTION OF THE NUMBER N OF EXTRATERRESTRIAL CIVILIZATIONS IN THE GALAXY The CLT may of course be extended to products of random variables upon taking the logs of both sides, just as we did in equation (3). it then follows that the exponent random variable, like Y in (6), tends to a normal random variable, and, as a consequence, it follows that the base random variable, like N in (6), tends to a lognormal random variable, 39 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 yegQjse". (49) This, upon inversion, yiclds the single root 47) =40)= ny). (50) On the other hand, differentiating (49) one gets g ()=e" where (50) was already used in the last step. The general transformation law (9) finally yields and g(x (vee™=y 6D fyO)= ECG lal) 7 per ne) (52) ROOM 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: Be L oh! oF den < 4) A>O, B=real.| (53) This follows immediately from the normalization condition of the Gaussian (7), that is fae? f Le 2 deal, (4) just upon expanding the square at the exponent and making the two replacements (we skip all steps) A= ! >0, 2 2 (55) B= a = real, = UNCLASSIFIED / /POTeOPPICTRE USE ORE
UNCLASSIFIED / / F@ROFFICii=U6E-ONn-™
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 ExtraTerrestrial Civilizations in
the Galaxy. In other words, 7 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 new on
the lognormal probability density function of N will
be written as
(nln bal
20° (m=0)) (56)
fyla)=
I
_. e
n \2xo
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 V
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, 1, 2,2... one
has
oe
(n*) =e e ? (57)
The relevant proof starts with the definition of the k-
th moment
One then transforms the above integral by
virlue of the substitution
Infr] =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
40
Upon setting k=0 into (56), the
normalization condition for f,, (2) follows
fiw (n)dn=1. (59)
Upon setting k=1 into (56), the important
mean value of the random variable N is found
oe
(N)=e"e2 |, (60)
Upon setting k =2 into (56), the mean valuc
of the square of the random variable N is found
(v?) ae gt (61)
The variance of N now follows from the last two
formulae:
oft =e ee a —1}. (62)
The square root of this is the important standard
deviation formuta for the N random variable
=
oy =e e 2 ¥e™ -11, (63)
The third moment is obtained upon setting
k =3 into (56)
vo
ao
(n*\=e%e (64)
Finally, upon setting & =4, the fourth moment
of N is found
(w*) met bo (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 y; by virtue of the
recursion formula (see ref. [8])
. fi-t .
Ke= uy > (ia) Ky byt (66)
UNCLASSIFIED / (POR -OFPEGIRI-UOErOREYUNCLASSIFIED / /P@R°OFFECHUSE-ONie
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:
‘ (67)
Ky=1y -3K, K,-K}
Ky = Hy -4K, Ky—-3K] -6Ky KP - KY.
These equations yield, respectively:
o
K,=ee?. (68)
Qu jo fet
Ky =e" e ( -1). (69)
94:
K,=ee? (70)
Ky= gee (« =) (.82" $307" 4667 +6) (71)
From these we derive the skewness
eee
(72)
la
and the kurtosis
"
430 4607 * +6)
Ky 4c 36°
set 4287 430% -6. (73)
(K,)°
Finally, we want to find the mode of the
lognormal probability densily function, i.e. the
abscissa of its peak. To do so, we must first
compute the derivative of the probability density
function fy(n) of equation (56), and then set it
equal to zero. This derivative is actually the
derivative of the ratio of two functions of x, as it
plainly appears from (57). Thus, let us set for a
moment
41
E(n)= {lela uy (14)
where “E™ stands for “exponent.” Upon
differentiating this, one gets
Eas Amba). (75)
But the lognormal probability density function (56),
by virtue of (74), now reads
een
fu))=—— (76)
20 n
So that its derivative is
dfer pistane {*) _ 1 -e BWE (n)-n-Le EW)
dr 220 we
_ =o fe (an +t) ay
V2n0 -
Setting this derivative equal to zero means setting
E (n)-n24+1=0 (78)
That is, upon replacing (75).
nfo] - 1) + 1=0. (79)
oO
Rearranging, this becomes
Infx]- +0? =0 (80)
and finally
2
Rude = Mea =O OF (81)
wade = peak
This is the most likely number of ExtraTerrestriat
Civilizations in the Galaxy.
How likely? To find the value of the probability
density function fy() corresponding to this
value of the mode, we must obviously replace (81)
into (56). After a few rearrangements, one then
gets
UNCLASSIFIED /G2@ReQEELGhAid SEO RiUNCLASSIFIED / / BG R=G-EEEGHAi GEO Abe Fy Oooie) = vet (82) 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 fy (n). Next to the mode, the median #7 (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 1 with 50% probability or a value higher than m with 50% probability again. In other words, the median # splits up our probability density in exactly two equally probable parts. Since the probability of occurrence of the random event equals the arca under its density curve (ic. the definite integral under its density curve) then the median « (of the lognormal distribution, in this case) is defined as the integral upper Limit at: Aina ee? Ht at | ] a¢? 1 [, fulo)dn = [ 7 Vino 2° (83) In order to find m, we may not differentiate (83) with respect to wt, since the “precise” factor 4 on the right would then disappear into a zero. On the contrary, we may try to perform the obvious substitution a (n(n) 4)? - 220 20° (84) into the integral (83) to reduce it to the following integral defining the error function erf(z) eef (x)= A fers (85) Then, after a few reductions that we skip for the suke of brevity, the full equation (83) is turned into 5te oof B-# #). > (86) 2a that is In(n)— 0 87) on fe “|; Since from the definition (85) one obviously has ert(0)=0, (87) becomes ln) a _ (88) “Qo whence finally [rection =m =e] (89) This is the median of the lognormal distribution af N. in other words, this is the number of ExtraTerrestrial civilizations in the Galaxy such that, with 50% probability the actual value af N will be lower than 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. Random variable N= number of communicating ET civilizations in Galaxy Probability distribution Lognormal tinea Probability density function fyj=t ee 2 zo “ n J220 Mean value Variance Standard deviation 42 UNCLASSIFIED / s2@8-@G ERISA UGEONE=—=
UNCLASSIFIED / /G02:05646bAirwGi-@ hai All the moments, ic. &-th moment Re (nt) =e" e 2 Mode (= abscissa of the lognormal peak) = — ptt y-F umde = Mpeak = &" € Value of the Mode Peak ! V2r0 = = Hg 2 Fi Ont) = se Mee Median (= fifty-fifty probability value for ¥) median = m1 =e" : yO go Skewness Ks == ( ' 2} - ee 5 (Ky)2 : Qo -y +30? +6e7 +6) Kurtosis Ky =e 42297 430° 6 (K2y Expression of gin terms of the lower (a;) and upper a= x (y)= ye [in(,)- )-1]J- a; [In ,)- 1 (b,) limits of the Drake uniform inpul random oT a i b,-a; in i variables Dj Expression of o? in terms of the lower (a,) and upper (6) limits of the Drake uniform input random variables D; u 2. a,b[in(b,)-In(a, JF a= Dei = - a a; iy got f-1 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 cquation (3) listed in Table 1. According to the CLT, the mean value ys to be inserted into the lognormal densily (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: =>) yeti Lalla )-1]] (90) —d; i i=l Upon replacing the 14 a; und 6, listed in Table 1 into (90), the following numeric mean value ji is found Similarly, to get the numeric variance o” one must resort to the last of equations (48) and to (33): Yo . v _ 4,6, [ino )— In(a; JP (92) is] fel (5, -a; y 43 yielding the following numeric variance o to be inserted into the lognormal pdf (56) whence the numeric standard deviation o (=aTmORT] os 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 in Table 1). We now like to point out the most important statistical propertics of this lognormal pdf: 1) Mean Vaiue of N. This is given by cquation (60) with yzand o given by (91) and (94), respectively: UNCLASSIFIED / /+@2-@5EiGIrUGEONE
UNCLASSIFIED / s-OR-@FFEGH=EEONEY (N) =e" ¢ 2 = 4589559 |, (98) | 9, In other words, there are 4590 ET Civilizations in the Galaxy according the Central Limit Theorem of Statistics with the inputs of Table 1. This number 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 lognormal distribution is given by (62) and turns out to be a huge number: oy =e e™ (7 - i}e 125328623. (96) 3) Standard deviation of N. The standard deviation of the lognormal distribution is given by (63) and turns oul to be: o oy see t ve" -1 = 11195 | (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: Mode = Mew =e" e@ 7 = 250 | (98) ampde This is well shown in Figure 4: the made 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. 44 5) Median of N. The median (= fifty-fifty abscissa, spliting the pdf in two exactly equi-probable parts) of the lognormal distribution of N is given by (89), and has the numeric value: In 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 luborious Fourier transform calculations in the complex domain, and 2) The lognormal distribution (56) with numeric ys and o given by (91) and (94) respectively. We sce 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 af the many facets. Similarly it happens for natural log of N, i.e. the random variable ¥ of (5), that is plotted in Figure 5 both in its normal curve version (thin curve) and in ils numeric version, obtained via Fourier transforms and already shown in Figure 2. The conclusion is simple: from now on we shall discard forever the numeric calculations and we'll stick only to the equations derived by virtue of the CLT, ie. ta the lognormal (56) and its consequences. UNCLASSIFIED / /FOROPPCTRE USE ONES
UNCLASSIFIED / (P@RsOFEEGHEUSEONEY PROBABILITY DENSITY FUNCTION OF N aio? LEN Ww L Prob. density function of N Y wh 0 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) Al the end of Section 3.3. ina purely numeric way and without resorting to the CLT at all (thick curve) and 2) Analytically by using the CLT and the relevant lognormal approximation (thin curve). PROBABILITY DENSITY FUNCTION OF Y=In(N) OS o4 / ) | f Probability density function of Y O22 / Ot 4 MS 0 i 2 3 4 5 6 7 8 9 10 ML 12 Independent variable ¥ = In(N) Figure 5. Comparing the two probability density functions of the random variable Y=In(N) found: 1) Al the end of Scction 3.3. in a purely numeric way and withoul resorting lo the CLT at all (thick curve) and 2) Analytically by using the CLT and the relevant normal (Gaussian) approximation (thin Gaussian curve). 8 DISTANCE OF THE NEAREST EXTRATERRESTRIAL CIVILIZATION AS A PROBABILITY DISTRIBUTION As an application of the Statistical Drake Equation developed in the previous sections of this paper, we now want to consider the problem of estimating the distance of the ExtraTerrestrial Civilization nearest to us in the Galaxy. In all Astrobiology textbooks (sce, for instance, ref. [10]) 45 and in several web sites, the solution to this problem is reported with only slight differences in the mathematical proofs among the various authors. In the first of the coming two sections (section 7.1) we derive the expression for this “ET_Distanee” (as we like to denote it} in the classical, non- probabilistic way: in other words, this is the classical,. deterministic derivation. In the second section (7.2) we provide the probabilistic derivation, arising from our Statistical Drake UNCLASSIFIED / /POR°OPPECEAEUGE-Ohire
UNCLASSIFIED /sh@8-OERIGIAR SEOs
Equation, of the corresponding probability density
function fir pistame(") : here r is the distance
between us and the nearest ET civilization
assumed as the independent variable of its own
probability density function. The ensuing sections
previde more mathematical details about this
Frrpistane(*) 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,
Reutayys 18 about 50,000 ly.
2) The thickness of the Galactic Disk at half-way
from its center, digugys is about 16,000 ly.
Then
3) The volume of the Galaxy may be
approximated as the volume of the
corresponding cylinder, i.e.
2
Vecataw = Reatay ht (100)
4) Now consider the sphere around us having a
radius r. The volume of such as sphere is
4 _/( ET_Distance
Vour_ Sphere = od
a
} (161)
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:
46
Veatasy _ Your sphere ; (102)
N 1
That is, upon replacing bath (100) and (101) into
(102):
$a Pounce)
2
TReatarh 37 es (103)
N | , ;
the only unknown in the last equation is
ET_Distance, and so we may solve for it, thus
getting the:
(AVERAGE) DISTANCE BETWEEN ANY PAIR
OF NEIGHBOURING CIVILIZATIONS IN
THE GALAXY
ET_Distance = ¥en" = © (104)
where the positive constant C is defined by
C= ae Réatacy Heiatayy 28845. light yeurs|, (105)
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 DENSITY FUNCTION FOR
ET_DISTANCE
The probability density function (pdf) yielding
the distance of the ET Civilization nearest to us in
the Galaxy and presented in this section, was
discovered by this author on September 5", 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 / /E@R:@EEIGHrUSEONEYUNCLASSIFIED / AF@R-OFEEGHiHUSE Oh
Let us go back to equation (104). Since N is
now a random variable (obeying the lognormal
distribution), it follows that the ET_Distance must
be a random variable as well. Hence it must have
some unknown probability density function that
we denote by
Fir ustane (> (106)
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
| Nox (107)
ET_distance > y
into the transformation law (8) for random
variables. As a consequence, (1{)4) takes form
yegiyeeaex. (108)
Var
In order to find the unknown probability density
Ferpistane(7) , 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
pare
a ()=-Sex a (110)
Thus, the relevant absolute value reads
47
4
a
4
=f x (1)
\e'(x] =
Cc
I y
3
Upon replacing (111) into (9), we then find
4
. ce -_cfety3_efe,_»*
le (nator delist) sil=| =4).
3 3 [y 3 [iy 3c
(112)
This is the denominator of (9). The numerator
simply is the lognormal probability density
function (56) where the old independent variable
must now be re-written in terms of the new
independent variable y by virtue of (109). By
doing so, we finally arrive at the new probability
density function fy (y)
ist
:
, - . 13
Varo am)
Fert aistame =
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:
\
f (j=3. te
J ET_Distane r Jie *
: (tl4)
holding for r20.
STATISTICAL PROPERTIES OF THIS
DISTRIBUTION
UNCLASSIFIED / 46 2-@>REGHirUGEO hieUNCLASSIFIED / APOPeOPPreIAE USP UnET
We now want to study this probability
distribution in detail. Our next questions are:
1) 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, ie.
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 = 0,1,2,3,... 20)
(E_Distance* ) = i‘ ré. FET_Distme (r) dr
=Che Fe 3B, (115)
To prove this result, one first transforms the above
integral by virtue of the substitution
3
HS] a. (116)
r
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
at 2 =
et
KE!
(Et_Distance’) = ¢* ete lo 17
Upon selling k=0 into (117), the
normalization condition for fet_pistane(7) follows
Yer Distane (1) dr = 1. (118)
48
Upon setting k =1 into (117), the important
mean value of the random variable ET_Distance
is found
woe
({ET_Distance) =Ce 4 ¢'*|, (119)
Upon setting k =2 into (117), the mean value of
the square of the random variable ET_Distance is
found
-" "s
(ET_Distance? ) =Ce xs ee, (120)
The variance of ET_Distance now follows from
the last two formulae with a few reductions:
OFT Distme = (Er. _Distance *) - (ET_Distance)”
2 oe o
aCe Mee eo a1]. U2H
So, the variance of ET_Distance is
2 12
Far visune =Coe * e*be% -tlL (122)
The square rool of this is the important
standard deviation of the ET_Distance random
variable
Hole
Fetvisune =Ce Ae! Ve? -11 (123)
The third moment is obtained upon setting
k =3 into (117)
a
2
(ET_Distance*) =Cie%e (124)
Finally, upon setting k =4 into (117), the fourth
moment of ET_Distance is found
4 83
(ET_Distance*) =C*e ar (125)
UNCLASSIFIED / A2@Re@EELGHAL=UGEeGaiaeUNCLASSIFIED / /POToOPEECEAE USE ONE
Our next goal is to find the cumulants of the
ET_Distance. In principle, we could compute all
the cumulants K; from the generic ith moment
#, by virtue of the recursion formula (see ref. [8])
. eet :
K=4->, (2) Ky Hank: (126)
ket
In practice, however, here we shall confine
ourselves to the computation of the first 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:
K=K
Ky = 1,-K
Ky = My -3K, Ky Kj}
Ky = ft, -4K, Ky-3K} -6K, Kj -K}.
(127)
These equations yield, respectively:
K,=Ce el, (128)
Ky,=CPe Fe%le® -1]. (129)
oe So o
K,=Cre*le? -3e #4206 |. (130)
Ky= (3)
sul &o? fic? 4c* o 26°
=Cle Fle’ -4e" -3e % +12 ¢3 -be ®
From these we derive the skewness
Ky
{K,)2
49
e Sat ot
ele? -3¢ 8 +266
8a" so” dc? ca 20°
of -4¢% -3e% +12¢? “6
-- (132)
and the kurtosis
4 =e % +2¢343¢% -6. (133)
(K,F
Next we want to find the mode of this
distribution, i-e. the abscissa of its peak. To do so,
we must first compute the derivative of the
probability density function fyy pistune (7) 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
Gag
Ajo (134)
where “E” stands for “exponent,” Upon
differentiating,
one gets
[SF afent. (135)
But the probability density function (113) now
reads
30 eH
Aictvisune (r) = Vine .
(136)
So that its derivative is
afer pistine (7) 3
dr 20 ‘ r?
UNCLASSIFIED / /P@?-OFFEGH=USEONEYUNCLASSIFIED / s£@R-@ERIGH Ain GEOR
(137)
Setting this derivative equal to zero means setting
E{r)-r+1=0 (138)
That is, upon replacing (135) into (138), we get
V {i
o
Xu
[S]-u) cabrereo (139)
7
Rearranging, this becomes
cl
{eS -1 +07 =0 (140)
re
that is
cl ;
-a0] +3y+0° =0 (141)
Pe
whence
jon a 2
mf S| 37 9 (142)
and finally
no
Frode = "peak =Ce Fe ? |. (143)
This is the most likely ET_Distance from Earth.
How likely ?
To find the value of the probability density
function fist piste (7) Corresponding to this value
of the mode, we must obviously replace Q) into ().
After a few rearrangements, which we skip for the
sake of brevity. one gets
Peak Value of fs pisane = ficr_pistume Conde)
2
o
3 ‘
= ced g)8,
Cv2a20
50
». (144)
This is the peak height in the pdf fey wissume(?)-
Next lo the mode, the median m (ref. [9]) is one
more statistical number uscd to characterize any
probability distribution, It is defined as the
independent variable abscissa im such that a
realization of the random variable will take up a
valuc lower than m with 50% probability or a value
higher than m with 50% probability again. In other
words, the median #7 splits up our probability
density in exactly two equally probable parts. Since
the probability of occurrence of the random event
equals the arca under its density curve (ic. the
definite integral under its density curve) then the
median a (of the lognormal distribution, in this
case) is defined as the integral upper limit x7:
oH
Fer victane ("er -+ (145)
0
Upon replacing (113), this becomes
"3 | a |
2 =-, 146
[ 5 (46)
6 aro
In order to finda, we may not differentiate (146)
with respect to m, since the “precise” factor % on the
right would then disappear into a zero, On the
contrary, we may Uy to perform the obvious
substitution
CG 2
? r
en cre (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 Icave to the reader as
an exercise, the full cquation (145), defining the
median, is turned into the corresponding equation
involving the error function erf(x) as defined by (85):
UNCLASSIFIED / /PO@?TeOPPICEREUSEOnEPUNCLASSIFIED / /#@feO@FFSG@iieU6E-ORieem
Random variable
ET_Distance between any two neighboring ET
Civilizations in Galaxy assuming they are UNIFORMLY
distributed throughout the whole Galaxy volume.
Probability distribution
Unnamed (Paul Davies suggested “Maccone distribution”)
Probability density function
Jet_vistane(")
{Defining the positive numeric constant C)
~ 2 .
C= i 6 RGanay Neutay w= 28845. light years
ne
{ET_Distunce) =Ce tel8
Mean value
ay of es
Variance Oey pistune =Ce Fe fe% =1
Standard deviati 3 Yeo
Standard deviation =Ce? 9
Ser pistme =Ce *2e'* Ve? -1
ee
All the moments, i.c. X-th moment (ET_Dis tance* ) =C*e 3¢@ I
tt oe
Mode (= abscissa of the probability density function Fide =Fpvak =C@ Fe ?
peak)
Peak Value of fer _pistane =
Value of the Mode Peak 3 noe
= firr_pistne Uinde) = Cline ret elf
H
Median (= fifty-fifty probability value for
ET_Distance)
median =i = Ce 3
a Sa? oy)
eer -3e!8 yet]
Skewness Kae
3 3
(K,)2 ta? sa? so? e 2a? \2
Cle ® -4¢°% -3¢% +12¢3 -6¢?
See
Kurtosis A-=¢% 4+2¢343¢% -6
(K,)
Expression of rin terms of the lower (¢;) and upper a= vy) - yale) 1-a,{in(a,)-1]
(b;) limits of the Drake uniform input random q aan. ; b,- a;
ip is
variables D;
Expression of o” in terms of the lower (a,) and upper
(b) limits of the Drake uniform input random
_,_ahlin(b,)- tne, SP
2 (o,-4,F
fol
variables D;
Table 3. Summary of the properties of the probability distribution that applies to the random variable ET_Distance
yielding the (average) distance hetween any two neighboring communicating civilizations in the Galaxy.
Si
UNCLASSIFIED / / F@R-@EEEGId= SSE ONEYUNCLASSIFIED / /#@R-@5EIGE-USEONR ===
m 1
—+erf =— (148)
2 J206 2
that is
In] ~, |-#
wa
er] | ——=—_|=0 (149)
2o
Since from the definition (147) one obviously has
ert(O)=0, (149) yields
This is the median of the lognormal distribution of
N. in other words, this is the number of
&xtraTerrestrial 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.
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
In this section we provide a numerical
example of the analytic calculations carried on so
far.
ce
ae Consider the Drake Equation values reported
ia 0 (150) in Table 1. Then, the graph of the corresponding
; id probability density function of the nearest
whence finally ET_Distance, fer pistane("), is shown in Figure 6.
“
median =1=Ce 4], (151)
503-102" DIST ANCE OF NEAREST ET_CIVILIZA TION
563-107
= asic
3
=
& aa f iN
8
3 \
5 22510”
=
4 IN
21340" in
a
0 S500 1000) 1500-2000
2500 3000 «3500 «4000 «4500 = S000
ET_Distance from Earth (light years)
Figure 6. This is the probability of finding the nearest ExtraTerrestrial Civilization at the distance r from
Earth (in light years) if the valucs assumed in the Drake Equation are those shown in Table |. The relevant
probability density function fy7 pisane(*) 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
52
UNCLASSIFIED / /S@R-GERIGSUS=OnS ==UNCLASSIFIED / / PO?teOPPECEREUSE-OnE fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR SETHI, 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 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 Ae Frode = "pen =C@ Xe 9 &1933 light years). (152) hod: = Spe This is the MOST LIKELY VALUE of the distance at which we can expect to find the nearest ExtraTerrestrial civilization. Tt is not, however, the mean value of the probability distribution (113) for fysrpistawe(?)- In fact, the probability density (113) 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 aoe Tuean_vatuc = CC selk x 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 2676 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 ExtraTerrestrials 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) oe s Or pistine =CE +8 2? —1 = 1309 light years}, 154) More precisely, this is the so called 1-sigma (distance) level. Probability theory then shows that the nearest ExtraTerrestrial civilization is expected to be located within this range, i.e. within the two distances of (2670-1309) = 136] light years and (2670+1309) = 3979 light years, with probability 53 given by the integral of fez pistane(7) taken in between these two lower and upper limits, that is: 397 9lightyears i) Set vistane (Par = 0.75 = 75% (155) 136 Nightyears 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 |. If we change those input values, then all the numbers change again. 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. 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 “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 UNCLASSIFIED / / POR OPPIRCIAE UST UNTY
UNCLASSIFIED/ / E@R-@GEIGHinwUGE-BiiES 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 first “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 hengur and privilege to present his results at the SETI Institute on April 11", 2008, in front of an audience 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 8", 2008. He also would like to thank Dan Werthimer and his School of SETI young experts for keeping alive the 54 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 Humans in the Galaxy. REFERENCES [1] http:/en.wikipedia.org/wiki/Drake_equation [2] hup://cn.wikipedia.org/wiki/SET] (3] http://en.wikipedia.org/wiki/Astrobiology [4] http://en.wikipedia.org/wiki/Frank Drake [5] Athanasios Papoulis and S. Unnikrishna Pillai, “Probability, Random Variables and Stochastic Processes”, Fourth Edition, Tata McGraw-Hill, New Delhi, 2002. ISBN 0-07-048658-1. (6] http://en.wikipedia.ore/wiki/Gamma_distribution 17] hup://en.wikipedia.org/wiki/Central_limit_thvore m [8] hup://en.wikipedia.org/wiki/Cumulants [9] hup-//en.wikipedia.org/wiki/Median [10]|Jeffrey Bennett and Seth Shostak, “Life in the Universe”, Second Edition, Pearson - Addison- Wesley, San Francisco, 2007, ISBN 0-8053- 4753-4. Sec in particular page 404. UNCLASSIFIED / / PO? OFFICE USEONEY
UNCLASSIFIED / (FP @fCOFFEGHirlGa-Obia= 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 1°, 2008, at the 59" International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29% thru October 3, 2008. 55 UNCLASSIFIED / (FP@?e @PFEGHEUSsE OnE