The statistical
Drake equation and A.M.Lyapunov theorem in problem of search for extraterrestrial
intelligence, part I Claudio Maccone SETI Permanent Study Group, The study is
connected with problems of mathematical modelling in the theory of Search
for Extraterrestrial Intelligence [1-13]. In this area at present it is known
the Frank
Drake equation. But it is very important to extend this model with using
brilliant theorem of statistics: Central Limit Theorem in form of Alexander
Lyapunov (or in form of Jarl Waldemar Lindeberg). In work the generalized model is
developed in form of the statistical Drake equation. This article is prepared
on materials of paper (IV IAA International Symposium, July, 2009, The Statistical Drake Equation The Drake equation (1961) is a mathematical way to spell out the seven most important events in the history of the Solar System that led from the Sun birth up to our communicating civilization. Exactly the same chain of seven events is supposed to have occurred anywhere else in the Galaxy, with the results that the product of these seven numbers yields the total number of extra-terrestrial civilizations communicating in the Galaxy right now. This is the classical Drake equation. It does not permit, however, any error bars to be taken into account for each of the seven input factors. The author of this paper thus sought to transform the classical Drake equation into a statistical product of seven (or more) input random variables, each of which is assigned with its mean value and standard deviation, and each of which is supposed to be uniformly distributed since the uniform distribution has the largest uncertainty (entropy). Then, the number N of communicating extraterrestrial civilizations becomes a random variable also. This author then proved that N follows the lognormal distribution whatever the distributions of the input random variables might be, if one just lets the number of input random variables tend to infinity (in practice, more than 5 input terms already yield a nearly perfect lognormal distribution). This discovery has profound consequences for SETI. For instance, the average distance in between two nearby communicating civilizations in the Galaxy became a new random variable whose probability density function was discovered by this author also (Maccone distribution). The final consequences of all this for SETI are synthesized in the Statistical Fermi Paradox, studied by this author in the last paper published herewith. According to a more detailed description, in this paper the statistical generalization of the Drake equation is attempted. 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 of the CLT, depending on the 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 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 practically very useful, of course. 3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighbouring 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 variables 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) 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 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. References 1.
http://en.wikipedia.org/wiki/Drake_equation 2.
http://en.wikipedia.org/wiki/SETI 3.
http://en.wikipedia.org/wiki/Astrobiology 4.
http://en.wikipedia.org/wiki/Frank_Drake 5.
Athanasios Papoulis and S. Unnikrishna Pillai,
"Probability, Random Variables and Stochastic Processes", Fourth Edition, Tata
McGraw-Hill, 6.
http://en.wikipedia.org/wiki/Gamma_distribution 7.
http://en.wikipedia.org/wiki/Central_limit_theorem 8.
http://en.wikipedia.org/wiki/Cumulants 9.
http://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. See in particular page
404. 11.
F.Drake.
Intelligent life in space. 12.
P.C.W.Davies.
Space and time in the modern universe. 13.
C.Sagan,: Communication with extraterrestrial intelligence. The
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