Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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(b, -a, )(1 + g) (36)
Thus, the characteri.~tic function of the natural log
t~fthe Drake u11(fr1rm random variahle D; i.1· given by
3.3 STEP 7: FINDING THE
PROBABILITY DENSITY
FUNCTION OF N, BUT ONLY
NUMERICALLY NOT
ANALYTICALLY
(371
Having found the characteristic functions
<1\ (;) of the log~ of the ~even input random
variables D, . we can now immediately fmd the
characteri~tic function of the random variable Y =
ln(N) defined by (5). In fact. by virtue of (4), of the
well-known Fourier transform property stating that
"the Fourier transform of a convolution is the product
of the Fourier transforms··, and of (37). it
immediately follow~ that cDv(!;) equals the product
of the ~even (!Jr,(.;):
The next ~lcp is to invert thi~ Fourier tran~form in
order to get the probability density function of the
rnndom variable Y = ln(N). In other word~, we mu~t
rnmpute the following inverse Fourier tran~form
(39)
37
Thi~ author rcgrL:b that he was unabk to computc the
la~t integral analytically. He had to compute it
1111merically for the particular values of the 14 u, and
b, that follow from Table I and equations 17. The
result wa~ the probability density function for Y =
ln(N) plotted in the following Figure 2.
PROB. DEJ\SITY FU"ICTION OF Y=ln(N)
0.4
03
0.2
0.1
() ()
•
I
~
.2 3 4 :, fl 7 8 9 JD I l 12
lm.lcpcndcnt vm"iablc Y = In("! J
Figure 2. Probability density function of Y = ln(N)
rnmputed numerically by virtue of the integral (39).
The two "funny gaps" in the curve nre due to the
numeric limitation~ in the MathCad numeric ~olver
that the author used for thi~ numeric computation.
We are now just one more step from finding the
probability den~ity of N. the number of
Extra Terrestrial Civilizations in the Galaxy predicted
by our Stati~tical Drake e4uation (]). The point here
i~ to transfer from the probability dcn~ity function of
Y to that of N, knowing that Y = ln(N), or
alternatively, that N=exp(YJ, a~ ~tatcd hy (6). We
must thus rc~ort to the tran~formation law of random
variable~ (9) by ~etting
y=g(x)=e'. (40)
Thi~, upon inversion. yields the single root
x, (y) C x(,) C In(,). (41)
On the other hand. differentiating (40) one gets
(42)
where (41) wa~ already med in the la~t ~tep. The
genernl trnn~formation law (9) finally yield~
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