Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

UNCLASSIFIED//F8"1 8ffll!lit.l! l!llili e,112~·
In the sequel of this paper we ~hall denote the
indcpcmknl iariabk uf lhc: lognormal distribution
(47) by a lower ca~e letter n to remind the reader that
corresponding random variable N 1s the po~itive
integer number of ExtraTerrestrial Civilization~ in
the Galaxy. In other words, 11 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. our~elve~. In conclu~ion, from now on
the Iognormal probability density function of 2\' will
be written as
fv(n)= _! ·-c;---
1-,
n ,(2.Jra
(n ~ (]) (56)
Having so said. we now turn to the statistical
properties of the log:normal distribution (55). i.e. to
the statistical properties that describe the number N
of Extra Terrestrial Civilizations 111 the Galaxy.
Our first goal 1s to prove an equation yielding all
thc momenb of the l(]gnormal <li~trihution (56), that
i~. for every non-negative integer k - 0, l, 2,. (me
has
(57)
The relevant proof ~tart~ with the definition of the k-
th moment
(N')= rnk·I..v(n)dn
40
(1nr11]-p)'
=I:n" 11.&a·e J,,
One then transform~ the above integral by
virtue of the sub\titution
ln[n]=:. (58)
The new integral in z is then seen to
reduce to the Ciau\sian integral (53)
(WC skip all \lCP\ here) and (57)
follow\
Upon setting k = 0 into (56).
normalization condition for f,v(n) follows
f~l.v(n)d11 = 1.
J,,
the
(59)
Upon setting k = 1 into (56), the important
mean value of the randam variable N is found
(60)
Upon setting k = 2 into (56). the mean value
of the square of the random variable N is found
(61)
The variance af N now follows from the last two
fornmlae:
(62)
The square root of this is the important .~ta11dard
deviation formula for the N random variable
(63)
The third moment is obtained upon setting
k = 3 into (56)
(64)
Finally, upon setting k = 4, the fou1th moment
of N is found
(65)
Our next goal is to find the cumulants of N. In
principle. we could compute all the cumulants K,
from the generic i-th moment p; by virtue of the
recursion formula (see ref. [8])
(66)
UNCLASSIFIED/ ;'flHl 8ffl@IAI!: l!l!iili 8Hl!:l/