Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

UNCLASSIFIED/ /FOR GI I tetlllt ~:!I! 8HLY
So, let us take the natural log~ of both ~ides of the
Stati~tical Drake equation (3) and change it intu a
~um:
It is now convenient to introduce eight new (positive)
random variables defined as follows:
/Y 0 h,(N)
[Y, c!n(D,) ;c!, .. J (5)
Upon inversion, the first equation of (5) yields the
important equation, that will he used in the ~cqucl
'N =e .
We are now ready to take STEP THREE.
(6)
STEP 3: THE TRANSFORMATION LAW
OF RANDOM VARIABLES
So far we did not mention nt all the problem:
"which probability di~tribution ~hnll we attach to
each of the seven (positive) random variables D, '!'.
It is not casy to answer thi~ qucstion because wc
do not ha,c thc lca~t scientific clue tu what
probability distributions fit at best tu cach of the
~even points li~ted in Section I.
Yet, at lea~t one trivial error must be avoided:
clniming that each of those seven random variables
must have a Gau~~ian (i.e. normal) di~tribution. In
fact, the Gaussian distribution, having the well-
known bell-~haped probability density function
(7)
has its independent variabk _v ranging between ---<:e
and -7~ and so it can apply to a real random variahlc
Y only, and never t(] pm·itive random variahlc~ like
those in the ~tati~tical Drake equntion (3). Period.
Searching again for probability den~ity functions
thnt repre~ent po~itive random variable~. nn obvious
choice would be the gammn di~tributions (see, for
in~tance, ref. [6]). However, we di~cnrded this choice
too because of a different rea~on: please keep in mmd
that, according to (5). once we selected a particular
32
type of probability density function (pdf) for the Inst
~cven uf cquatium (5), thcn we mu~t compute thc
(new and different) pdf of the logs of such random
variable~. And the pdf of these log~ certainly 1s not
gamma-type any more.
It is high tune now to remind the reader of a
certain theorem that 1s proved in probability courses,
hut, unfortunately, doe~ not seem to have a specific
name. It i~ the tra11sformatio11 law (so we ~hall call
it, ~ee, for instance, ref. [5 Jl allowing u~ to compute
the pdf of a certain new random variahle Y that is a
known functi(]n Y = g(X) (If another random
variable X havmg a known pdf. In other word~, if the
pdf fx (x) of a certain random variahle X i~ known,
then the pdf fr (_r) of the new random variable Y.
related to X hy the functional rclation~hip
(8)
can be calculated according: to this rule:
I) First invert the corresponding: non-probab1li~t1c
equation _\·=g(x) and denote by x,(_r) the
various real roots resulting from the this
mvers1011.
2)
3)
Second, take notice whether these real roots may
be either finitely- or infinitely-many. according
to the nature of the function _I'= g(x).
Third, the probability density function of Y i~
then given by the (finite or infinite) sum
(9)
where the summation extends to all roots x, (.r) and
lx'(x,(r)~ is thc ab~olutc valuc of thc fir~l
derivative of g(x) where the i-th root x, (_\') has
bccn rcplaccd imtead of .r.
Since we must use thi~ transformation law to tran~fer
from the D; to the Yi = ln(DJ, it i~ clear that we
need to start from n D, pdf that is ns ~irnple a~
po~~ihle. The gamma pdf is not responding to this
need hecau~e the analytic express1on of the
transformed pdf is very complicated (or, at lea~t, it
looked so to this author in the fir~t instance). Aho,
the gamma di~trihution ha~ two free parameters in it,
and this ··complicates" ib application to the variuu~
mcanings of the Drake equation. In conclusion, we
discarded the gamma distributiun~ and confined
UNCLASSIFIED/ .(COP OCCJCJ0I I 155 ONI X