Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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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 Chuir, St.:11 Permanent Study Group, lnternational Academy of lls!rvnu11lics
Address: Via Martorelfi, 43 - Torino (Turin) 10155 - Italy
URL: hllp://www.macconc.com/ - E-mail: clmaccon@'libcro.it
ABSTRACT. We provide the statistical generalization of the Drake e4uation.
From a simple product of ~even po~itive 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 loo~e 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:
I) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follmvs the
LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the
ordinary Nin the Drake equation. The ~tandard 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 ,vords, the CLT "translates" into our statistical Drake equation by allmving an arbitrary
probability distrihution for each factor. This is both physically realistic and practically very useful, of
cour~e.
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 /\NY positive numher 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
re~ults in the fields of Aqrohiology and SETI.
Finally. a practical example is given of how our ~tatistical 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
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