Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

UNCLASSIFIED/,'f81il 8ffll1Als WliEii SUllslf
ourselves to the simpler uniform di~tribution instead,
as shown in the nesl ~ection.
4. STEP 4, ASSUMING THE EASIEST
INPUT DISTRIBUTION FOR EACH Di:
THE UNIFORM DISTRIBUTION
Let u.~ now suppose that each of the .~even D1 i.~
di.\·tributed UNIFORMLY in the interval ranging
from the fower limit a1 :C: U to the upper limit
h, 2 ui.
This is the same as saying th.it the probability
den~ity function of each of the seven Drake random
variables D, has the equation
funifrirmo{x)=--
1- withOS:a; S:xS:h; (10)
- ' b; -a,
as it follows at once from the normalization condition
I
,,'funiforrn_D, (x)dx= I.
"
(I))
Let us now consider the mean value of such
uniform D, defined by
I,, I I''(unifonn_D,)= 'xfun,R,rrnD(x)dx=--- 'xdx
", - ' b;-a, a,
By W(1n.b (a~ it is intuitively ohviou~): the mean
value of the uniform diMribution ~imply is the mean
of the lowl:r plu~ upper limit of thl: iariablc range
( ) u-+b.
unifom1_D 1 = -'-2-•. (]2)
In order to find the variance (If the uniform
distrihuti(]n, we first need finding the ~ec(]nd moment
(uniform_o, 2
)= f"x 2 /~nlfnun_o,(x)dx
"·
• •h,' - a;
3(b, -a 1 )
33
_ (b, -a 1 )(a,1 +a;b, +b,2) _ a,1 +a;b, +b,2
3(h,-a,) 3
The second moment of the uniform di~trihution is
thus
• •
(
.
1
. o') a,-+a,b;+b;-
um om1 , =- 3 (13)
From (12 and (13) we may now denve the variance
of the uniform distribution
crJ,ufom_D, = ( uniform_D ,2 )-(uniform_D,) 2
a} +a,b, +b,2
(a 1 +h1 )2 (h, -a,)2
3 4 12 (14)
Upon taking the ~quare root of both sides of (14), we
finally obtain the standard deviation of the 111tiform
distribution:
h, - (I;
2fj (15)
We now wi~h t(] perform a calculation that is
mathematically trivial, hut rather unexpected from
the intuitive poinl of view, and very important for our
applications to lhc stalistical Drakl: equation. Jusl
cun~idl:r the lwu ~imultancuu~ equaliuns (12) and
(15)
1
a +h
(unifom1_D,) = ~
h -a
(,Ullli<>rlLL_IJ, = ~fi/.
Upon inverting: this trivial linear system, one fmds
fa, = (unifonn_D; )- fi O"umt<nm o.
1h, = (un ifonn_D;) + fi O"LLnikm,,_D,
( 16)
(17)
Thi~ 1s of paramount importance for our application
the Stati~tical Drake equation ina~much a~ it shows
that:
if one (scientifically) assigns the mean value a1td
standard deviation of a certain Drake random
variable D;, then the lower and 11pper limits of the
relevant uniform distribution are given by the two
equations (17), respectively.
UNCLASSIFIED//:r:aA: 8FFifiil1l1ls W~Eii a•11slf