Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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Thi~ probability density function fN(Y) wa~
computed numerically hy using (4]) and the numeric
curve given hy (39), and the result i~ shown in figure
3.
PROBABILITY Dl:.:-,/SITY /-UNlTIO:-,i OF N
Z .t-10-4
~
'· .....
00 1()00 2000 3()0() 4000
:-,/ = :-lumber of ET Ci"~izatiom in Gala~-
Figure 3. Tht: numeril' (and nut analytic) probability
density function curve fN(y) of the number N of
Extra Terrestrial Civili1:ations in the Galaxy accmding
to the Statistical Drake cquati(]n (]). We sec that the
curve peak (i.e. the mode) i~ very close to l(]W values
(If N, hut the tail on the right is high, meaning that the
n::sulting mean valut: (N) is of thc order of
thou~ands.
We now want to compute the mean value (N)
of the probability density (43). Clearly, it is given by
(N)~ f,·r,C,•),1,. (44)
"
This integral too was computed numerically, and the
result was a perfect match with N=3500 of (22), that
"
(N) = 3499.99880 177509 +O.axxxx.112 4914686i (45)
Note that thi~ re~ult wa~ computed numerically in the
complex domain because of the Fourier transform~.
and that the real part 1s virtually 3500 (as expected)
while the imaginary part i~ virtually zero because of
the rounding errors. So, this result is excellent, and
proves that the theory presented so far 1s
mathematically correct.
Finally we want to consider the standard
deviation. Thi~ aho had to he computed numerically,
resulting in
r,N = ]95].42910 143389 +0.0'.XfX:O:B 28CXXl'i8i . (46)
38
This standard deviation, higher than the mean value,
implies that N might range in bctwn::n O and 7453.
This completes our study of the probability
den~ity function of N if the seven uniform Drake
input random variable D, have the mean values and
~tandard deviations li~ted Ill Table 1.
We conclude that, unfortunately, eve11 under the
.~implifying a.\·.rnmption.~ that the D, be unijOrmly
di.\·tributed, it is impo.\·.\·ible to solve the full problem
analytically, since all calculation.~ beyond equation
(38) had to be perJOrmed numerically,
This is no good,
Shall Wl: thus luosc faith, and dt:clarc •'impmsihlc"
the task of finding an analytic cxprc~sion for the
probability density functiun f..v(Y) •J
Rather surprisingly, the answer is "no", and there
is indeed a way out of this dead-end, as we shall sec
in the next sccti(]n.
5. THE CENTRAL LIMIT THEOREM (CLT)
OF STATISTICS
Indeed there is a good, approximating analytical
expression for f.,v (_v) , and this 1s the following
l«gnormal probability demity .function
f.v (y,_u,a) = _!_ ·-,,;---
1-c
Y ✓ 2,-va
(y <". 0). (47)
To understand why, we must resort to what is
perhaps the most beautiful theorem of Statistics:
the Central Limit Theorem (abbreviated CLT).
Historically, the CLT was in fact proven first in
190 I by the Russian mathematician Alexandr
Lyapunov ( 1857-1918), and later ( 1920) by the
Finnish mathematician far] Waldemar Lindeberg
( 1876-1932) nnder weaker conditiom. These
conditions are certainly fulfilled in the context of
the Drake: equation hccausc of the "reality" of the:
astronomy, biology and sociology involved with it,
and we are not going to discuss thi~ point any
further here. A good, synthetic description of the
Central Limit Theorem (CL T) of Statistic~ is found
at the Wikipedia site (ref. [7J) to which the reader
is referred for more dctails, such as the cquatiom
for the Lyapunov and the Lindeberg conditions.
making the theorem ''rigorously" valid.
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