Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

UNCLASSIFIED//FOAt OFFI&l11J.k laUiEii SU.b'J.1
,, a-
All the moments, i.e. k-th mumrnt (Nk)=f'" e '
Mude (= abscissa 11f the lognmmal peak) " ~a -,,,,, -c-
,mJe - pe;ik -
I a
Value nf the Mode Peak ' ,
f-\, (nirrnlc) = ,Jz;; ' -, -
21T a
Median(= fifty-fifty prohability value for N) n-edian = m = el-'
----"--'-- ~ (,"' I 2) -/w -10"'
Skewnes~ ,, ,,
;
(, a _ I)'(,,a + 3e""' - )'(K, ), +6ecr- +6
Kurtu~is K4 ➔ a'
--~,
(K, )' + 2 e'"' + 3 e 2"' -6
Expression of pin terms of the lower (o,) and upper _L, ( \_L, h,[ln(h;)-1]-a;[h,(a,)-1]
(h,) limib 11f the Drake uniform input random jl- Y,;-
,-1 ,-1 h, -a,
variabks D,
Exprc~sion of a' in terms of the lower (a,J and upper
, , a,h, [h,(b, )- in(a, )]'
rr' ~ Lrrf ~ LI-
(h,) limits of the Drake uniform input random (b, -a, )2,_, i-1
variables D,
Table 2. Summary of the propertie~ of the lognormal di~tnbution that applies to the random variable N = number of
ET rnm1mmicating civilizatiom in the Galaxy.
\Ve want t(] cnmplcte this sectinn ahout the
lognormal prnhahility demity functi(]n (56) hy
finding nut it~ 11umerie value.~ for the inputs tn the
Stati~tical Drake e4uatinn (]) listed in Table I.
According tn the CLT, the mean value μ to he
in~crtcd into the lognurrnal den~ity (56) i~ given
(according to the ~econd equation (48)) by the ~um of
all the mean value~ (Y;). that is. by virtue of (31 ). by:
Upun replacing the 14 a, and b1 li~ted in Tabk I
into (90), the following numeric mean value p i~
found
(91)
Similarly, to get the numeric variance a' one
must resort to the la~t of equations (48) and to (33):
,
cr2 = Lcrf (92)
i=I
43
yielding the follmving numeric variance a 2 tn he
in~erted into the lognormal pdf (56)
la',,, 1.9387251 (93)
whence the 11umeri£· standard deviation rr
(94)
Upon replacing the~e two numeric value~ (84)
and (86) into the lognormal pdf (56), the latter 1s
perfectly determined. It i~ plotted in Figure 4
hereafter as the thin curve.
In other words, Figure 4 shows the lognormal
distributio,1 for the number 2\' of ExtraTerrestrial
Civilizations in the Galaxy derived from the Central
Limit Theorem as applied ta the Drake equatia1t
(with tlte input data listed in Table 1).
We now like to point out the most important
~tati~tical prnpcrtics nf thi~ lognormal pdf:
I) Mean Value of N. Thi~ i~ given hy cquati(]n (60)
with /I and rr given hy (91) and (94). respectively:
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