Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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Let us go back to equation (104). Since N is
now a random variable (obeying the lognormal
distribution), it follows that the ET _Distance mmt
be a random variable as well. Hence it must have
some unknown probability demity function that
we denote by
(106)
where r i~ the new independent variable of such a
probability di~tribution (it is denoted by r to
remind the reader that it expresses the three-
dimensional radial distance separating us from the
nearest ET civilization in a foll spherical symmetry
of the space around us).
The question then is: what is the unknown
probability distribution (106) of the ET _Distance?
We can answer this question upon making the two
formal substitutions
{
N ➔ x
El"_diqancc ---'I- _,,
(107)
into the transformation law (8) for random
variables. As a consequence, (104) takes form
C
r=~(x)=-=C·x -'.. . v;; ( 11)8)
In order to rind the unknown probability density
fcT_D1.,t,uHc(r), we now to apply the rule (9) to
(108). First. notice that (108), when inverted to
yield the various roots x, (y), yields a single real
root only
( 11)9)
Then, the summation in (9) reduces to one term
only.
Second. differentiating (108) one finds
, C -
~ (x)=-___::___·X 1
3
Thus, the relevant absolute value reads
47
(110)
X
C -
·X _1.
Upon replacing ( 111) into (9), we then find
(111 J
... (112)
Thi~ is the denominator of (9). The numerator
simply i~ the lognormal probability density
function (56) where the old independent variable x
must now be re-written in terms of the new
independent variable _l' by virtue of ( 109). By
doing so, we finally an-ive at the new probability
density function Ir Cr)
Rearranging and replacing y by r. the J'inal form
JS:
I. ( ) 3 I
. ETJC,,"""' r =-• ~ e
- r ,J2HCY
( II 3)
Now, just replace C in (113) by virtue of (105).
Then:
We have discovered the probability density
function yielding the probability of finding the
nearest ExtraTerrestrial Civilization in the
Galaxy in the spherical .~hell betwee11 the
di.~tances rand r+dr.from Earth:
( I 14)
holding for r ::C: 0.
STATISTICAL PROPERTIES OF THIS
DISTRIBUTION
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