Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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Put in loose terms, the CIX states that, if one
has a sum of random variables even NOT
identically di.\'lrihuted, thi.1' .mm tend.~ ta a normal
di.\·trihution when the number of term.\' making up
the sum tends to i1ifinity. AJ.rn, the normal
di.\·trihution mean value i.1· the .mm rd· the mean
value.1· af the addend random variable.\', and the
normal di.1·tributian variana i.~ the .mm of the
varianu.~ of the addend random variable.1·.
Let us now write down the equations of the CLT
in the form needed to apply it to our Statistical Drake
equation (3). The idea i~ to apply the CLT to the sum
of random variahles given hy (4) and (5) whatever
their prahahility di.1·trihutiom can po.l'.l'ibly he. In
(Jthcr word~. the CLT applied to the Statistical Drake
equation (3 J leads immediately to the following three
equations:
1J The sum of the (arhitrarily di~trihuted)
independent random variables Y, makes up
thc new random variable Y.
2) The sum of their mean values makcs up thc
new mean value uf Y.
3) The sum uf thcir ,ariances makes up the
new variance of Y.
In equations:
'_,
,
O'i = La{
'_,
(48)
This completes our synthetic description of the CLT
for sums of random variables.
6. THE LOGNORMAL DISTRIBTION IS
THE DISTRIBUTION OF THE NUMBER
N OF EXTRATERRESTRIAL
CIVILIZATIONS IN THE GALAXY
The CLT may of course be extended to products
of random variables upon taking the logs of both
sides, just as we did in equation (3). It then follows
that the exponent random variable, like Y i11 (6),
tends to a normal random variable, and, as a
consequence, it follows that the base random
variable, like Nin (6), tends to a log11ormal random
variable.
39
To understand this fact better in mathematical
terms consider again of the transformation law (9) uf
random variables. The question is: what is the
probability density function of the random variable N
in equation (6). th.it is, what 1s the probability density
function of the lognormal d1stribut1on? To find 1t. set
(49)
This, upun inversion. yiclds the .~ingle root
On the other hand. differentiating (49) (me get~
where (50) was already used in the last step. The
general transformation law (9) finally yields
Therefore, replacing the probability density on the
right by virtue of the well-known normal (or
Gaussian) distribution given by equation (7), the
lognormal distribution of equation (47) is found, and
the derivation of the log-normal distribution from the
normal distribution is proved.
In view of future calculations, it is also useful to
point out the so-called "Gaussian integral''. that is:
,,
1
-, -,1-,'H,d H '' A 0e e x= -·e , >,
--, A B =real. (53)
This follows immediately from the normali1:ation
condition of the Gau~~ian (7). that is
i.,-p)'
la' dx =I. (54)
just upon expandmg the square at the exponent and
makmg the two replacements (we skip all steps)
l I
11=--,, >0,
2a-
l'
B = a" = real.
(55)
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