AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

UNCLASSIFIED/ /POil OFFl@IAI:: WSli 0Nk¥ Defense Intelligence Reference Document Acquisition Threat Support 11 March 2010 !COD: 1 December 2009 An Introduction to the Statistical Drake Equation UNCLASSIFIED/ /FOA OFFICIO L: 1PiF ON! Y UNCLASSIFIED//P81t 8FFIEIAL Y&E &ttbl/ An Introduction to the Statistical Drake Equation Prepared by: Acquisition Support Division (DW0-3) Defense Warning Office Directorate for Analysis Defense Intelligence Agency Author: AAP Person 80 Administrative Note COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized. This product is one in a series of advanced technology reports produced in FY 2009 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapon System Applications (AAWSA) Program. Comments or questions pertaining to this document should be addressed t o !AAP Person 1 ~ AAWSA Program Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington, DC 20340-5100. UNCLASSIFIED//F8R 8FFl61tlib Wlilli Qtlb¥ ii UNCLASSIFIED//POil OFFl@IAL WSli 0Nk¥ Contents 1. Introduction .......................................................................................................iv 2. The Key Question: How Far are They ? .............................................................. 4 3. Computing N By Virtue of the Drake Equation (1961) ........................................ 7 4. The Drake Equation is Over-Simplified ............................................................. 10 5. The Statistical Drake Equation ......................................................................... 11 6. Solving the Statistical Drake Equation By Virtue of the Central Limit Theorem (CLT) of Statistics .......................................................... , .................................... 13 7. An Example Explaining the Statistical Drake Equation ..................................... 15 8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical Drake Equation................................................................................................. 18 9. The "Data Enrichment Principle" as the Best CLT Consequence Upon the Statistical Drake Equation (Any Number of Factors Allowed) ........................... 23 10. Conclusions .................................................................................................... 23 Appendix A: Proof of Shannon's 1948 Theorem Stating That the Uniform Distribution is the "Most Uncertain" One Over a Finite Range of Appendix B: Original Text of the Author's Paper #IAC-08-A4.1.4 Entitled the Values ............................................................................................. 25 Statistical Drake Equation ............................................................... 28 References ........................................................................................................... 55 UNCLASSIFIED/fEOR OFFICIO! 11SF ON! X iii UNCLASSIFIED//POil OFFl@IAI:: WSli 0Nk¥ An Introduction to the Statistical Drake Equation 1. Introduction SETI (an acronym for "Search for Extraterrestrial Intelligence") is a relatively new branch of scientific research, having begun only in 1959. Its goal is to ascertain whether alien civilizations exist in the universe, how far from us they exist, and possibly how much more advanced than us they may be. As of 2009, the only physical tools we know that could help us get in touch with aliens are the electromagnetic waves an alien civilization could emit and we could detect. This forces us to use the largest radiotelescopes on Earth for SETI research, because the higher our collecting area of electromagnetic radiation is, the higher our sensitivity is (that is, the farther in space we can probe). Yet, even by using the largest radiotelescopes on Earth (the 310-meter dish at Arecibo, for instance), we cannot search for aliens beyond, say, a few hundred light years away. This is a very, very small amount of space around us within our galaxy, the Milky Way, that is about 100,000 light years in diameter. Thus, current SETI can cover only a very tiny fraction of the galaxy, and it is not surprising that in the past 50 years of SETI searches, NO extraterrestrial civilization was discovered. Quite simply, we did not get far enough! This demands the construction of much more powerful and radically new radiotelescopes. Rather than big and heavy metal dishes, whose mechanical problems hamper SETI research too much, we are now turning to "software radiotelescopes," where a large number of small dishes (ATA = Allen Telescope Array, and ALMA= Atacama Large Millimeter/submillimeter Array) or even just of simple dipoles (LOFAR = Low Frequency Array) using state-of­ the-art electronics and very-high-speed computing can outperform the classical radiotelescopes in many regards. The final dream in this field is the SKA ( = Square Kilometer Array), currently being designed and expected to be completed around 2020. 2. The Key Question: How Far are They? But still, the key question remains: how far are they? Or, more correctly, how far do we expect the NEAREST extraterrestrial civilization to be from 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 rad io 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, R Gala.1y , is about 50,000 ly. UNCLASSIFIED/ /FOR OFFI&IJ.k Wili Ql'II.¥ iv UNCLASSIFIED//fOR OFFI&IAk Wlii QPU,¥ The thickness of the Galactic Disk at half-way from its center, h a " 1t'-'Y ' is about 16,000 ly. The volume of the galaxy may then be approximated as the volume of the corresponding cylinder, i.e. (1) Now consider the sphere around us having a radius r. The volume of such a sphere is 4 ( Ef Distance) 3 Vo ur _ Sphere = 3n - (2) 2 In the last equation, we had to divide the distance "ET _Distance" between ourselves and the nearest ET civilization by 2 because we are now going to make the unwarranted assumption that all ET civilizations are equally spaced from each other in the galaxy! This is a crazy assumption, clearly, and should be replaced by more scientifically-grounded assumptions as soon as we know more about our Galactic Neighborhood. At the moment, however, this is the best guess that we can make, and so we shall take it for granted, although we are aware that this is a weak point in the reasoning. Furthermore, let us denote by N the total number of civilizations now living in the galaxy, including ourselves. Of course, this number N is unknown. We only know that N ~ 1 since one civilization does at least exist! Having thus assumed that ET civilizations are UNIFORMLY SPACED IN THE GALAXY, we can then write down the proportion: Va a/a.,y Vo ,,r_Sphere -- = -~- (3) N That is, upon replacing both (1) and (2) into (3): 3 4 ( Ef- Dis lance ) 2 - i'l" _ ___ i'l" RGa/axy h = 3 2 (4) N 1 The last equation contains two unknowns: N and ET_Distance, and so we don't know which one it is better to solve for. However, we may suppose that, by resorting to the (rather uncertain) knowledge that we have about the Evolution of the galaxy through the last 10 billion years or so, we might somehow compute an approximate value for N. Then, we may solve (4) for ET_Distance thus obtaining the (AVERAGE) DISTANCE BETWEEN ANY PAIR OF NEIGHBORING CIVILIZATIONS IN THE GALAXY (DISTANCE LAW) UNCLASSIFIED/;'FQA QFFICiIPL. !Iii 011! Y 5 UNCLASSIFIED//fOR OFFI&IAk Wlii QPU,¥ 3 6 R2 h Ef ff (N) Ca!a,y _ ,stance = VN C (5) VN where the positive constant C is defined by C =V6 R ~ala.,y h ca/axy ,., 28845 light years . (6) Equations (5) and (6) are the starting point to understand 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 N may not be smaller than 1, i.e., N ~ 1. Suppose then that we are alone in the galaxy, i.e., that N=l. Then the distance law (5) yields as distance to the nearest civilization from us just the constant C, i.e., 28,845 light years. 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=l000, i.e. there are about a thousand extraterrestrial communicating civilizations in the whole galaxy right now. Then the distance law (5) yields an average distance of 2,885 light 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. • Suppose finally that N=l00000O, i.e., there are a million communicating civilizations now in the galaxy. Then the distance law (5) yields an average 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 (1960-2010). In conclusion, interpolating the above three special cases of N, we may say that the distance law (5) yields t he following key diagram of the average ET distance vs. the assumed number of communicating civilizations, N, in the galaxy right now (Figure 1): UNCLASSIFIED/ J'FOA. OFlilCI0 I. P !SE ON! X 6 0 l \ ' "-­ ~ UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ Average DIST A CE of the nearest ET civilization vs. the ASSUMED NUMBffi of ET civilizations in the Gah Cl) 0::: <( UJ >­!­ :I: :i .!:: !'.l g ~ !'l= 200 175 150 125 100 750 500 25 0 0 0 I 00000 200000 300000 400000 500000 600000 700000 800000 900000 I 000000 ASS MED NUMBER of civiliz ations in the Galaxy (that is, Nin the Drake equation) Figure 1. DISTANCE LAW; i.e., the Average Distance (plot along the ver-1:ical axis in light years) Versus the NUMBER of Communicating Civilizations ASSUMED to Exist in the Galaxy Right Now 3. Computing N By Virtue of the Drake Equation (1961) In the previous section, the problem of finding how close the nearest ET civilization may be was "solved" by reducing it to the computation of N, the total number of extraterrestrial civilizations now existing in this galaxy. In this section the famous Drake equation is described, that was proposed back in 1961 by Frank Donald Drake (born 1930) to estimate the numerical value of N. We believe that no better introductory description of the Drake equations exists other than the one given by Carl Sagan in his 1983 book "Cosmos" (ref. [2]), in its turn based on the famous TV series "Cosmos." So, in this paragraph we report Carl Sagan's description of the Drake equation unabridged. "But is there anyone out there to talk to? With a third or a half a trillion stars in our Milky Way galaxy alone, could ours be the only one accompanied by an inhabited planet? How much more likely it is that technical civilizations are a 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 id ly at a star we call the Sun and entertaining, for just a moment, an outrageous speculation. UNCLASSIFIED//FOR OFFIEiIAk Wlii QPllsV 7 UNCLASSIFIED//fOR OFFI&IAk Wlii QPU,¥ It is very hard to be sure. There may be several impediments to the evolution of a technical civilization. Planets may be rarer than we think. Perhaps the origin of life is not so easy as our laboratory experiments suggest. Perhaps the evolution of advanced life forms is improbable. Or it may be that complex life forms evolve more readily, but intelligence and technical societies require an unlikely set of coincidences - just as the evolution of the human species depended on the demise of the dinosaurs and the ice­ age recession of the forests in whose trees our ancestors screeched and dimly wondered. Or perhaps civilizations arise repeatedly, inexorably, on innumerable planets in the Milky Way, but are generally unstable; so all but a tiny fraction are unable to survive their technology and succumb to greed and ignorance, pollution and nuclear war. It is possible to explore this great issue further and make a crude estimate of N, the number of advanced civilizations in the galaxy. We define an advanced civilization as one capable of radio astronomy. 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: • Ns, the number of stars in the Milky Way galaxy. • fp, the fraction of stars that have planetary systems. • ne, the number of planets in a given system that are ecologically suitable for life. • fl, the fraction of otherwise suitable planets on which life actually arises. • fi, the fraction of inhabited planets on which an intelligent form of life evolves. • fc, the fraction of planets inhabited by intelligent beings on which a communicative technical civilization develops. • fl, the fraction of planetary lifetime graced by a technical civilization. Written out, the equation reads N =Ns •jjJ •ne •fl ·ft ·Jc· fL (7) All of the f's are fractions, having values between Oand 1; they will pare down the large value of Ns. To derive N 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. UNCLASSIFIED//FQA QFFICiIPL. !Iii 011! Y 8 UNCLASSIFIED//fOR OFFI&IAk Wlii QPU,¥ We know Ns, the number of stars in the Milky Way galaxy, fairly well, by careful counts of stars in a small but representative region of the sky. It is a few hundred billion; some recent estimates place it at 4 x 1011 . 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 1011 (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 1011 . 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 fl~ 1/3, implying a total number of planets in the Milky Way on which life has arisen at least once as Ns fp ne fl~ 1 x 1011 , a hundred billion inhabited worlds. That in itself is a remarkable conclusion. But we are not yet finished. The choices of fi 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 fix fc = 1/100, meaning that only 1 per cent of planets on wh ich life arises actually produce a technical civilization. This estimate represents some middle ground among the varying scientific options. Some think that the equivalent of the step from the emergence of trilobites to the domestication of fire goes like a shot in all planetary systems; others 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 1 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 51 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//POil Offl@IAL W&i 9Nls¥ 9 UNCLASSIFIED//FOR OFFIEiIAk Wlii QPU,¥ these factors together, we find Ns fp ne fl fi fc ~ 1 x 109, 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 Ns fp ne fl fi fc fl ~10, and, at a given time there would be only a tiny smattering, a handful, a pitiful few technical civilizations in the galaxy, the steady state number maintained as emerging societies replace those recently self-immolated. The number N might be even as small as 1 if civilizations tend to destroy themselves soon after reaching a technological phase; there might be no one for us to talk with but ourselves. And that we do but poorly. Civilizations would take billions of years of tortuous evolution, and then snuff themselves out in an instant of unforgivable neglect. But consider the alternative, the prospect that at least some civilizations learn to live with technology; that the contradictions posed by the vagaries of past brain evolution are consciously resolved and do not lead to self destruction; or that, even if major disturbances occur, they are reveres in the subsequent billions of years of biological evolution. Such societies might live to a prosperous old age, their lifetimes measured perhaps on geological or stellar evolutionary time scales. If 1 percent of civilizations can survive technological adolescence, take the proper fork at this critical historical branch point and achieve maturity, then fl ~ 1/100, N ~ 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 N, that various authors obtained by different assumptions about the astronomy, the biology and the sociology implied by the Drake equation, may range from a few tens (in the pessimist's view) to some UNCLASSIFIED//FOR OFFIEiIAk Wlii QPII.¥ 10 UNCLASSIFIED//FOR OFFIEil.t.k W~i Q~lls¥ million or even billions in the optimist's opinion. A lot of uncertainty is thus affecting our knowledge of N as of 2010. In all cases, however, the final result about N has always been a sheer number, i.e., a positive integer number ranging from 1 to millions or billions. This is precisely the aspect of the Drake equation that 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 1st , 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29th thru October 3rd, 2008. That paper is attached herewith as Appendix B. Newcomers to SETI and to the Drake equation, however, may find that paper too difficult to be understood mathematically at a first reading. Thus, I shall now 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., Ns, the number of stars in the Milky Way galaxy. Astronomers tell us that approximately there should be about 350 millions stars in the galaxy. Of course, nobody has counted (or even seen in the photographic plates) all the stars in the galaxy! There are too many practical difficulties preventing us from doing so: just to name one, the dust clouds that don't allow us to see even the Galactic Bulge (i.e. the central region of the galaxy) in the visible light (although we may "see it" at radio frequencies like the famous neutral hydrogen line at 1420 MHz). So, it doesn't make any sense to say that Ns = 350 x 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 (SO millions, in the above example). By doing so, we have made a great step ahead: we have abandoned the too-simplistic equation (7) and replaced it by something more sophisticated and scientifically more serious: the STATISTICAL Drake equation. In other words, we have transformed the classical and simplistic Drake equation (7) into an advanced statistical tool for the investigation of a host of facts hardly known to us in detail. In other words still: • We replace each independent variable in (7) by a RANDOM VARIABLE, labeled D, (from Drake). • We assume that the MEAN VALUE of each D, is the same numerical value previously attributed to the corresponding independent variable in (7). • But now we also ADD A STANDARD DEVIATION cr0; on each side of the mean value, that is provided by the knowledge gathered by scientists in each discipline encompassed by each D,. UNCLASSIFIED//FQA QFFICilOL. !!ii ONI X 11 UNCLASSIFIED//fOR OFFI&IAk Wlii QPU,¥ Having so done, the next question is: How can we find out the PROBABILITY DISTRIBUTION for each D; ? For instance, shall that be a Gaussian, or what? This is a difficult question, for nobody knows, for instance, the probability distribution of the number of stars in the galaxy, not to mention the probability distribution of the other six variables in the Drake equation (7). There is a brilliant way to get around this difficulty, though. We start by excluding the Gaussian because each variable in the Drake equation is a POSITIVE (or, more precisely, a non-negative) random variable, while the Gaussian applies to REAL random variables only. So, the Gaussian is out. Then, one might consider the large class of well-studied and positive probability densities called "the gamma distributions," but it is then unclear why one should adopt the gamma distributions and not any other. The solution to this apparent conundrum comes from Shannon's Information Theory and a theorem that he proved in 1948: "The probability distribution having maximum entropy ( = uncertainty) over any FINITE range of real values is the UNIFORM distribution over that range," This is proven in Appendix A of the present document. So, at this point, we assume that each of the seven D; in (7) is a UNIFORM random variable, whose mean value and standard deviation is known by the scientists working in the respective field (let it be astronomy, or biology, or sociology). Notice that, for such a uniform distribution, the knowledge of the mean value Po; and of the standard deviation u 0, automatically determines the RANGE of that random variable in between its lower (called a;) and upper (called b;) limits: in fact these limits are given by the equations (8) (the "surprising" factor ✓3 in the above equations comes from the definitions of mean value and standard deviation: please see equations (12), (15) and (17) in Appendix B for the relevant proof). So the uniform distribution of each random variable D; is perfectly determined by its mean value and standard deviation, and so are all its other properties. The next problem is the following: OK, since we now know everything about each uniformly distributed D;, what is the probability distribution of N, given that N is the product (7) of all the D1? In other words, not only do we want to find the analytical expression of the probability density function of N, but we also want to relate its mean value f-lN to all mean values µ of the D;, and its standard deviation to all standard deviations of the D;. , u N u , 0 0 12 UNCLASSIFIED/ fFOR: QFFICiIO L. lalii ODIL.¥ UNCLASSIFIED//FOR OFFI&I.t.k W~i Q~lk¥ This is a difficult problem. It occupied the author's mind for no less than about ten years (1997-2007). It is actually an ANALYTICALLY UNSOLVABLE problem, in that, to the best of this author's knowledge, it is IMPOSSIBLE to find an analytic expression for any FINITE PRODUCT of uniform random variablesD; . This result is proven in Sections 2 thru 3.3 of Appendix B (unfortunately!) . 6. Solving the Statistical Drake Equation By Virtue of the Central Limit Theorem (CLT) of Statistics The solution to the problem of finding the analytical expression for the probability density function of N in the statistical Drake equation was found by this author in September 2007. The key steps are the following: • Take the natural logs of both sides of the statistical Drake equation (7). This changes the product into a sum. • The mean values and standard deviations of the logs of the random variables D; may all be expressed analytically in terms of the mean values and standard deviations of the D; . • Recall the Central Limit Theorem (CLT) of statistics, stating that (loosely speaking) if you have a SUM of independent random variables, each of which is ARBITRARILY DISTRIBUTED (hence, also including uniformly distributed), then, when the number of terms in the sum increases indefinitely (i.e. for a sum of random variables infinitely long) .. . the SUM RANDOM VARIABLE TENDS TO A GAUSSIAN. • Thus, the natural log of N tends to a Gaussian. • Thus, N tends to the LOGNORMAL DISTRIBUTION. • The mean value and standard deviations of this lognormal distribution of N may all be expressed analytically in terms of the mean values and standard deviations of the logs of the D, already found previously. This result is fundamental. All the relevant equations are summarized in the following Table 1. This table is actually the same as Table 2 of the author's original paper IAC-08-A4.1.4, entitled "The Statistical Drake Equation" and presented by him at the International Astronautical Congress (IAC) held in Glasgow, UK, on October l5t, 2008. This original paper is reproduced in Appendix B. To sum up, not only is it found that N approaches the completely known lognormal distribution for an INFINITY of factors in the statistical Drake equation (7), but the way is paved to further applications by removing the condition that the number of terms in the product (7) must be FINITE. UNCLASSIFIED//FOR 8FFI&l.t.k Wii QNk¥ 13 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ This possibility of ADDING ANY NUMBER OF FACTORS IN THE DRAKE EQUATION (7) was not envisaged, of course, by Frank Drake back in 1961, when "summarizing" the evolution of life in the galaxy in SEVEN simple STEPS. But today, the number of factors in the Drake equation should already be increased: for instance, there is no mention in the original Drake equation of the possibility that asteroidal impacts might destroy the life on Earth at any time, and this is because the demise of the dinosaurs at the K/T impact had not been yet understood by scientists in 1961, and was so only in 1980! In practice, the number of factors should INCREASE as much as necessary in order to get better and better estimates of N as long as our scientific knowledge increases. This is called the "Data Enrichment Principle" and believe should be the next important goal in the study of the statistical Drake equation. Finally, a numerical example explaining how the statistical Drake equation works in the practice will be given in the next section. UNCLASSIFIED/ /rOR orr1e1At HSE OrtLY 14 UNCLASSIFIED/ /FOR OFFIEiIAk Wlii QPU,¥ Table 1. Summary of the Properties of the Lognormal Distribution That Applies to the Random Variable N = Number of ET Communicating Civilizations in the Galaxy Random variable N = number of communicating ET civilizations in galaxy Probability distribution Loqnormal Probability density function (1n(11h,J' 1 1 -~ (n ;::-: 0) n 27!<:T JN (n)= - · & e a' Mean value - (N )= eµe 2 Variance a1 = e2µ ea' (ea' - I) a' a N = eµ e 2 .Jea2 - 1 Standard deviation All the moments, i.e. k-th moment 2 a' k ·­2 (N k)= ekµ e _ _ ~I - a-2 Mode (= abscissa of the log normal peak) n nnde = npeak - e e a' f ( ) Value of the Mode Peak 1 -p 2 N n lTl)dc = & ·e ·e 2,r (l Median (= fifty-fifty probability value for rred ian =m =e" N) Skewness K ( , ) e-6pe-3a' _ 3_ = ea + 2 (K4)¾ ., lne3a2 + 3e2a' +6ea' +61 k' ­ K ,, ., ., "} Kurtosis _ 4_ = e4a- + 2 e3a- + 3 e-u- - 6 (K2)2 Expression of 1-1 in terms of the lower (a;) µ = ± (Y;) =± b;[in(b;) - 1]- a;[in(a;)-1] and upper (b;) limits of the Drake i=I i= I b; - a; uniform input random variables 0 ; Expression of <:T 2 in terms of the lower (a;) 7 7 a;b;[ln (b; )- in(a;)f a 2 =La~ = L l­ and upper (b;) limits of the Drake (b;-a} i= I i=I uniform input random variables 0 ; 7. An Example Explaining the Statistical Drake Equation To understand how things work in practice for the statistical Drake equation, please consider the following table 2. It is made up of three columns: • The first column on the left lists the seven input sheer numbers that also become • The mean values (middle column). • Finally the last column on the right lists the seven input standard deviations. UNCLASSIFIED/1re1t OFFl@IAL WSE 9Nk¥ 15 UNCLASSIFIED//FOR OFFIEil.t.k W&i Q~lls¥ The bottom line is the classical Drake equation (7). We see that, for this particular set of seven inputs, the classical Drake equation (i.e. the product of the seven numbers) yields a total of 3500 communicating extraterrestrial civilizations existing in the galaxy right now. rs := 350 -109 s := s 0 10 µfp := fp fp := 100 ofp := 100 l ne := 1 µne := ne crne := ­/3 0 fl := - µfl := fl crfl := ~ 100 100 fi :=~ µfi := fi on := ~ 100 100 0 fc:= - µfc := fc crfc := ~ 100 100 fL := 10000 crtL := 1000 µfl. := tL 1010 1010 _ 1 := Ks-fp-ne-fl-fi.fc.ff, = 3500 Table 2. Input Values (i.e. mean values and standard deviations) for the Seven Drake Uniform Random Variables Di . The first column on the left lists the seven input sheer numbers that also become the mean values (middle column). Finally the last column on the right lists the seven input standard deviations. The bottom line is the classical Drake equation (7). The statistical Drake equation, however, provides a much more articulated answer than just the above sheer number N = 3500. In fact, a MathCad code written by this author and capable of performing all 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 N the lognormal distribution (thin curve) plotted in Figure 2. We see immediately that the peak of this thin curve (i.e. the mode) falls at about n rmde;;;; npeak = eµ e-a' ""250 (this is equation (99) of Appendix B), while the median (fifty­ fifty value splitting the lognormal density in two parts with equal undergoing areas) falls at about nm,dian;;;; eµ ""1740 . These seem to be smaller values than N = 3500 provided by the classical Drake equations, but it's a wrong impression due to a poor "intuitive" understanding of what statistics is! In fact, neither the mode nor the median are the " really important" values: the really important value for N is the MEAN VALUE! Now if you look at 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 UNCLASSIFIED// POR OPPICll<L ti.!! er~LV 16 UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ (]'' surprised to find out that it equals (N) = e11 e2 ::::: 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 extraterrestriaI civiIization ! PROBABILITY DENSITY FUNCTION OF N 1000 2000 3000 4000 N = Number of ET Civilizations in Galaxy Figure 2. Comparing the Two Probability Density Functions of the Random Variable N Found (1) Without Resorting to the CLT at All (thick curve) and (2) Using the CLT and the Relevant Lognormal Approximation (thin curve). Even more so our hopes are increased when we go on to consider the standard deviation associated with the mean value 4590. In fact, the standard deviation is given ,,., by equation (97) of Appendix B. This yields er"' = e11 e2 ✓e" 2 -1 =11195 and so the expected number of N may actually be even much higher than the 4590 provided by the mean value alone! The "upper limit of the one-sigma confidence interval" (as statisticians call it), i.e. the sum 4590+11195 = 15,785, yields a higher number still! (Note: the "lower limit of the one-sigma confidence interval is ZERO because the lognormal distribution is POSITIVE (or, more correctly, non-negative)). Finally, the reader should note that the thick curve depicted in Figure 2 is just the NUMERICAL solution of the statistical Drake equation for a FINITE number of 7 input factors. Figure 2 actually shows that this curve "is well interpolated" by the lognormal distribution (thin curve), i.e., by the neat analytical expression provided by the Central Limit Theorem for an INFINITE number of factors in the Drake equation. That is, in conclusion, Figure 2 visually shows that taking 7 factors or an infinity of factors "is almost the same thing" already for a value as small as 7. UNCLASSIFIED/ /FOil OFFI@IAb W&i QNI.¥ 17 UNCLASSIFIED//FOR OFFIEil.t.k Wii Q~lls¥ 8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical Drake Equation Having solved the statistical Drake equation by finding the lognormal distribution, we are now in a position to solve the ET-DISTANCE problem by resorting to statistics again, rather than just to the purely deterministic Distance Law (5), as we did in Section 2. This is "scientifically more serious" than just the purely deterministic Distance Law (5) inasmuch as the new statistical Distance Law will yield a PROBABILITY DENSITY for the Distance, with the relevant mean value and standard deviation. In other words, the Distance Law (5) itself becomes a random variable whose probability distribution, mean value and standard deviation must be computed by "replacing" into (5) the fact that N is now known to follow the lognormal distribution. This is mathematically described in detail in Section 7 of Appendix A. The important new result is the PROBABILITY DENSITY FOR THE DISTANCE, the equation of which is (9) holding for r ;,: O. This is equation (114) of Appendix B. Starting from this equation, the MEAN VALUE OF THE random variable ET_DISTANCE is computed as J.I cr2 (Er_Distance) =Ce3 e""jg (10) which is equation (119) of Appendix B, and finally the ET_DISTANCE STANDARD DEVIATION _!!. a2Kr2 - 3 18 9 CJ ET_Distanu: - Ce e e - l (11) which is equation ( 123) of Appendix B. Of course, all other descriptive statistical quantities, such as moments, cumulants etc. can be computed upon starting from the probability density (9), and the 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: _}!_ CTl r,11ea11 _ m lue =Ce 3 e 18 ;::: 2,670 light years (12) UNCLASSIFIED//FOR OFFIEil.t.k Wii QNls¥ 18 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ which is equation (153) of Appendix B, and I' (12 ~ a ET_Di stono, =Ce 3 e •8 ~ e9 - 1 a:: 1,309 light years (13) which is equation (154) of Appendix B. UNCLASSIFIED/ /FOR &FFIGIAk lalii ODIL:¥ 19 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Table 2. Summary of the Properties of the Probability Distribution That Applies to the Random Variable ET_Distance Yielding the (average} Distance Between Any Two Neighboring Communicating Civilizations in the Galaxy Random variable ET_Distance between any two neighboring ET civilizations in galaxy assuming they are UNIFORMLY distributed throughout the whole qalaxy volume. Probability distribution Unnamed Probability density function _(•{6 R[ata '.I l,Gola')]-µ J 1 3 I 2u2 fET_Distana,(r) =-;. ·Ji; CT ·e Numerical constant C related to the Milky Way size C =3 6 R 8alruy h Gala.,y "" 28,845 light years Mean value µ u2 (Er_Distance) =C e-3 e18 Variance 2 2 2 - 2 2 a -3p 9 [ a9 -l CTET_Distance - C e e e I Standard deviation 2 _ji_ er ✓ a - 3 18 9 _ CTET_Distance - C e e e l All the moments, i.e. k-th moment k2,u2 - k!!. (Er_Dis tan eek) =c k e 3 e 18 Mode (= abscissa of the log normal peak) _ji_ er 9 rnnde =rpeak = Ce 3 e Value of the Mode Peak Peak Value of fET_Distanu:(r) = a' !!. 3 - ·e 3 . e 18 =fET_Distana,Cr,rnde) = cfi; CT Median ( = fifty-fifty probability value for N) _ji_ ~dian =m =Ce 3 Skewness ,,., 5,,.2 ,,., ] e-µ [ e 2 - 3e 18 +2e 6 __!!i_ = 3 3 (K4 )2 8a2 5a2 4a2 2 a 2a2 ]2 C3 [ e9 -4e- 9 -3e9 +12 e3 - 6e- 9 Kurtosis 4 a 2 a' 2 a 2 - - K4 _ 9 +2e 3 +3e 9 -6 (K2)2 - e Expression of µin terms of the lower (ai) and upper (bi) limits of the Drake uniform input random variables Di µ= I (r;) = I b; [ln (b;)- 1]- a;[ln(a;)- 1] i=I ;~1 b; - a; Expression of CT 2 in terms of the lower (ai) and upper (bi) limits of the Drake uniform input random variables Di 7 7 a;b; [ln (b;)- ln(a;)f CT 2 = Io} =Ii (b; - a; )2 i=I i=I UNCLASSIFIED/ J'FQA QFFICiIPL. !Iii 011! Y 20 UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ UNCLASSIFIED//POil OFFI@IAb W&i 91'11.¥ 21 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ It is clarifying to draw the graph of the ET_Distance probability density (9): DISTANCE OF NEAREST Ef_CIVILIZA TION 5.63 ·10-20 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 ET _Distance from Earth (light years) Figure 3. The Probability of Finding the Nearest Extraterrestrial Civilization at the distance r From Earth (in light years) if the Values Assumed in the Drake Equation are Those Shown in Input Table 1. The relevant probability density function fET_Disian.,( r) is given by equation (9). Its mode (peak abscissa) equals 1933 light years, but its mean value is higher since the curve has a long tail on the right: the mean value equals in fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR SETI, inasmuch as the nearest ET galaxy civilization might lie at just 1 sigma = 2670-1309 = 1361 light years from us. From Figure 3 we see that the probability of finding extraterrestrials is practically zero up to a distance of about 500 light years from Earth. Then it starts increasing with the increasing distance from Earth, and reaches its maximum at _l!_ ~ r=de = r peak = Ce 3 e 9 :::: 1,933 light years. (14) This is the MOST LIKELY VALUE of the distance at which we can expect to find the nearest extraterrestrial civilization. It is not the mean value of the probability distribution (9) for fET_Distan.,( r). In fact, the probability density (9) has an infinite tail on the right, as clearly shown in Figure 3, and hence its mean value must be higher than its peak value. As given by (10) and (12), its ....!:!.. 0'2 mean value is r,,,en,,_mlue = Ce 3 e 18 c::,2670 light years. This is the MEAN (value of the) DISTANCE at which we can expect to find extraterrestrials. UNCLASSIFIED/ ;'FQA QFFICiIPL. !Iii 011! Y 22 UNCLASSIFIED//FOR OFFIEiIAk Wlii QPU,¥ After having found the above two distances (1933 and 2670 light years, respectively), the next natural question that arises is: "what is the range, back and forth around the mean value of the distance, within which we can expect to find 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), I' 0'2 ~ CTET_Distance = Ce 3 e •8 Ve 9 -1 ""1309 light years . 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) = 1361 light years and (2670+1309) = 3979 light years, with probability given by the integral of fET_oistance(r) taken in between these two lower and upper limits, that is: i 3979 1igh1 years fET Disiance (r) dr:::: 0.75 = 75 % (15) l36 1 ligbtycars ­ In plain words: with 75 percent probability, the nearest extraterrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by table 1. If we change those input values, then all the numbers change again, of course. 9. The "Data Enrichment Principle" as the Best CLT Consequence Upon the Statistical Drake Equation (Any Number of Factors Allowed) As a fitting climax to all the statistical equations developed so far, let us now state our "DATA ENRICHMENT PRINCIPLE." It simply states that "The Higher the Number of Factors in the Statistical Drake equation, The Better." Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the normal distribution when the number of terms in the sum (4) approaches infinity. And this is the case, indeed. 10. Conclusions We have sought to extend the classical Drake equation to let it encompass Statistics and Probability. This approach appears to pave the way to future, more profound investigations intended not only to associate "error bars" to each factor in the Drake equation, but especially to increase the number of factors themselves. In fact, this seems to be the only way to incorporate into the Drake equation more and more new scientific information as soon as it becomes available. In the long run, the Statistical Drake equation might just become a huge computer code, growing in size and especially in the depth of the scientific information it contains. It would thus be Humanity's first "Encyclopaedia Galactica." UNCLASSIFIED//FOR OFFIEiIAk Wlii QPlls¥ 23 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ Unfortunately, to extend the Drake equation to Statistics, it was necessary to use a mathematical apparatus that is more sophisticated than just the simple product of seven numbers. UNCLASSIFIED/ /FOR 8FFIGl.t.k W&i QPII.¥ 24 1 UNCLASSIFIED/ /FOR OFFIEil.t.k W~i Q~lls¥ Appendix A: Proof of Shannon's 1948 Theorem Stating That the Uniform Distribution is the "Most Uncertain" One Over a Finite Range of Values Information Theory was initiated by Claude Shannon (1916-2001) in his well-known 1948 two papers: Rqrutted mm commi:ms from B •1-S,'l•st Ii •ral Jour:!li, \ GI. 1 . pp. 9--!13 , 62J..,iS , July, Occober. I A Mathematical Theory of Communication By C. E SHANNO In this Appendix, we wish to draw attention to a couple of theorems that Shannon proves on pages 36 and 37 of his work, and read, respectively (note that Shannon omits the upper and lower limits of all integrals in the first theorem: they are minus infinity and plus infinity, respectively): 5. Letp(x) bea one-dimensionaldis :i ti The onno p(x) ginng ammcimumentropy 1:.ubjec-tto the c dition tha e dan:l devia on ofx be fixed at a i- G ian. To show this we must maximize H(x) = - fp x)logp(x)d <T- =j p x- dx and 1 =/ p x)dx as cons •. This requi~ . by calculu~ of variations maxunizing / [- p x)logp(x) >.p(x 1p(x)] dx. e condition or lus is - 1- logp(x •ng the constllllf5 to sat' the and UNCLASSIFIED/ /FOR: QFFIGI0ls Uii 0111 X 25 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ 7. Ifx is limited to a half line (p(x) = 0 for x <0) and the fin.t moment ofx is ihted at a: a = f p(x dx then the maximwn en opy OCCW1! when p(x) =,! - f.r/al a and 1s equal to logea. Now, we wish to point out that there is a third possible case, other than the two given by Shannon. This is the case when the probability density function p(x) is limited to a FINITE INTERVAL a::; x::; b. This is obviously the case with any physical POSITIVE random variable, such as a distance, or the number N of extraterrestrial communicating civilizations in the,". And it is easy to prove that for any such finite random variable the maximum entropy distribution is the UNIFORM distribution over a -5, x -5, b. Shannon did not bother to prove this simple theorem in his 1948 papers since he probably regarded it as too trivial. But we prefer to point out this theorem since, in the language of the statistical Drake equation, it sounds like: "Since we don't know what the probability distribution of any one of the Drake random variables D; is, it is safer to assume that each of them has the maximum possible entropy over a; -5,x-5, h;, i.e., that D; is 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; -5, x -5, b; . We then form the linear combination of the entropy integral plus the normalization condition for D; where i is a Lagrange multiplier. Performing the variation, one finds - Iogp(x)- 1+1 = 0 that is: p(x)= e ,1,- i . Applying the normalization condition (constraint) to the last expression for p(x) yields b, ( ) f.b; ,l,-1 2-1 f.b; ,1,-1 ( ) I = f. p x dx = e dx = e dx = e b; - a,. a1 a1 a1 that yields ,l,-1 1 e = -­ h; - a; UNCLASSIFIED/ /POlt Offl@IAL WSi 9Nk¥ 26 UNCLASSIFIED/ /FOR OFFI&IAk Wlii QPU,¥ and finally p(x) = - 1- with a; s x s b; b; - a; showing that the maximum-entropy probability distribution over any FINITE interval a; sxsb; is the UNIFORM distribution . UNCLASSIFIED/ /FOR OFFI&IAk Wlii QNk¥ 27 UNCLASSIFIED/ /FOR OFFIEil.t.k W&i Q~lls¥ Appendix B: Original Text of the Author's Paper #IAC-08­ A4.1.4 Titled the Statistical Drake Equation IAC-08-A4.1.4 THE STATISTICAL DRAKE EQUATION Claudio Maccone Co-Vice Chair, SETI Permanent Study Group, International Academy ofAstronautics Address: Via Martorelli, 43 - Torino (Turin) 10155 - Italy URL: http://www.maccone.com/ - E-mail: [email protected] ABSTRACT. We provide th 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 (CL T) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. u01mal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of th CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: l) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary Nin the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are found also. 2) The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practicalJy very useful, of course. 3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function , apparently previously unknown and dubbed "Maccone distribution" by Paul Davies. 4) DATA ENRICHMENT PRINCIPLE. It should be noticed that ANY positive number of random vai'iables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation we call the "Data Enrichment Principle", and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. Finally, a practical example is given of how our statistical Drake equation works numerically. We work out in detail the case where each of the seven random variables is uniformly distributed around its own mean value and has a given standard deviation. For instance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 350 billions with a standard deviation of (say) I billion. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows UNCLASSIFIED/; P"OR OP"P"lelltt li!I! er•tv 28 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ that the mean value of the lognormal random variable N is actually of the same order as the classical N given by the ordinary Drake equation, as one might expect from a good statistical generalization. 1. INTRODUCTION The Drake equation is a now famous result (see ref. [1] for the Wikipedia summary) in the fields of SETI (the Search for ExtraTetTestial Intelligence, see ref. [2]) and Astrobiology (see ref. [3]). Devised in l 960, the Drake equation was the first scientific attempt to estimate the number N of ExtraTerrestrial civilizations in the Galaxy with which we might come in contact. Frank D. Drake (see ref. [4]) proposed it as the product of seven factors: N=Ns-fp-ne·fl·fi·fc·fL. (1) Where: I) Ns is the estimated number of stars in our Galaxy. 2) fp is the fraction (= percentage) of such stars that have planets. 3) ne is the number "Earth-type" such planets around the given star; in other words, ne is number of planets, in a given stellar system, on which the chemical conditions exist for life to begin its course: they are "ready for life," 4) fl is fraction(= percentage) of such "ready for life" planets on which life actually starts and grows up (but not yet to the "intelligence" level). 5) fl is the fraction (= percentage) of such "planets with life fom1S" that actually evolve until some form of "intelligent civilization" emerges (like the first, historic human civilizations on Earth). 6) Jc is the fraction (= percentage) of such "planets with civilizations" where the civilizations evolve to the point of being able to communicate across the interstellar distances with other (at least) similarly evolved civilizations. As far as we know in 2008, this means that they must be aware of the Maxwell equations governing radio waves, as well as of computers and radioastronomy (at least). 7) fl is the fraction of galactic civilizations alive at the time when we, poor humans, attempt to pick up their radio signals (that they throw out into space just as we have done since l 900, when Marconi started the transatlantic transmissions). In other words, fl is the number of civilizations now transmitting and receiving, and this implies an estimate of"how long will a technological civilization live?" that nobody can make at the moment. Also, are they going to destroy themselves in a nuclear war, and thus live only a few decades of technological civilization? Or are they slowly becoming wiser, reject war, speak a single language (like English today), and merge into a single "nation", thus living in peace for ages? Or will robots take over one day making "flesh animals" disappear forever (the so-called "post-biological universe")? No one knows ... But let us go back to the Drake equation (1). In the fifty years of its existence, a number of suggestions have been put forward about the different numeric values of its seven factors. Of course, every different set of these seven input numbers yields a different value for N, and we can endlessly play that way. But we claim that these are like ... children plays! We claim the classical Drake equation (1), as we shall call it from now on to distinguish it from our statistical Drake equation to be introduced in the coming sections, well, the classical Drake equation is scientifically inadequate in one regard at least: it just handles sheer numbers and does not associate an error bar to each of its seven factors. At the very least, we want to associate an error bar to each D;. Well, we have thus reached STEP ONE in our improvement of the classical Drake equation: replace each sheer number by a probability distribution! The reader is now asked to look at the flow chart in the next page as a guide to this paper, please. 2. STEP 1: LETTING EACH FACTOR BECOME A RANDOM VARIABLE In this paper we adopt the notations of the great book "Probability, Random Variables and Stochastic Processes" by Athanasios Papoulis (1921-2002), now re-published as Papoulis-Pillai, UNCLASSIFIED/ ,'FOR: QFFIGI CL. Wii ODIL:¥ 29 UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥ ref. [5]. The advantage of this notation is that it makes a neat distinction between probabilistic ( or statistical: it's the same thing here) variables, always denoted by capitals, from non-probabilistic (or "detenninistic") variables, always denoted by lower-case letters. Adopting the Papoulis notation also is a tribute to him by this author, who was a Fulbright Grantee in the United States with him at the Polytechnic [nstitute (now Polytechnic University) of New York in the years 1977-78-79. We thus introduce seven new (positive) random variables D; ("D" from "Drake") defined as D1 = Ns D2 =fp D3 =ne D4 = fl (2) Ds =fl D6 =Jc D7 =fL so that our STATISTICAL Drake equation may be simply rewritten as (3) Of course, N now becomes a (positive) random variable too, having its own (positive) mean value and standard deviation. Just as each of the D; has its own (positive) mean value and standard deviation ... ... the natural question then arises: how are the seven mean values on the right related to the mean value on the left? ... and how are the seven standard deviations on the right related to the standard deviation on the left? Just take the next step ... 3. STEP 2: INTRODUCING LOGS TO CHANGE THE PRODUCT INTO A SUM Products of random variables are not easy to handle i