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

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UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥
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 ExtraTerrestriaJ Civilizations in
the Galaxy. In other words, n 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 110w 011
the log11or111al probability density f11nction of N will
be written as
(1n(11}--µ)2
1
1 -~
( )
(56)
n v2na
IN n =-· r;:;-
e
(n~ O)
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 N
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, I, 2, . . . one
has
(57)
The relevant proof starts with the definition of the k­
d1 moment
(111[11]-p)'
k
I
I
-~
= rn •~ •.Ji;a •e
dn
0
One then transforms the above integral by
virtue of the substitution
In[n]= 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
Upon
setting
k =0
into
(56),
the
normalization condition for f N (n) follows
Upon setting k =1 into (56), the important
mean value ofthe ra11dom variable N is found
(60)
Upon setting k =2 into (56), the mean value
of the square of the random variable N is found
/N2) -
211 2a2
\
-e e
.
(61)
The variance of N now follows from the last two
formulae:
(62)
The square root of this is the important standard
deviation f onnula for the N random variable
(63)
The third moment is obtained upon setting
k =3 into (56)
(64)
Finally, upon setting k =4, the fourth 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])
•
i- 1 (i - 1)
.
(66)
K; =A-L k-1 K k f.l11-k·
k=i
UNCLASSIFIED/ /rOR orrl@IAL HSE O,.LY
40