Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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The bottom line is the classical Drake equation (7). We see that, for this particular set
of seven inputs, the classical Drake equation (i.e. the product of the seven numbers)
yields a total of 3500 communicating extraterrestrial civilizations existing in the galaxy
right now.
"'~ C 350 10' ~t-..:s := Ks cr-..: s C 1 109
<o
fj, 100 pfj, C fj, 10
crfj, C
100
n, C 1 ,m, C n, crn, C
j}
'.'O
fl C p.fl -= fl
100
10
crfl C -
100
'°fi C pfi := fi
100
10
crfi C
100
20
fr C ~tfc C fr
100
10
otC C
100
10000
fL C pfL := fl.
10 11J
1000
crfL C
1010
-..:.=Xsl.pneflfifctl.. " - 3500
Table 2. Input Values (i.e. mean values and standard deviations) for the Seven Drake Uniform Random
Variables Di . The first column on the left lists the seven input sheer numbers that also become the mean values
(middle column). Finally the last column on the right lists the seven input standard deviations. The bottom line is
the classical Drake equation (7).
The statistical Drake equation, however, provides a much more articulated answer than
just the above sheer number N = 3500. In fact, a MathCad code written by this author
and capable of performing all the numerical calculations required by the statistical
Drake equation for a given set of seven input mean values plus seven input standard
deviations, yields for N the log normal distribution (thin curve) plotted in Figure 2. We
see immediately that the peak of this thin curve (i.e. the mode) falls at about
n,mJe = n1
wak = e1-1 e-cr "'250 (this is equation (99) of Appendix B), while the median (fifty-
fifty value splitting the lognormal density in two parts with equal undergoing areas) falls
at about nmxlLan = c1-1 "'1740 . These seem to be smaller values than N = 3500 provided by
the classical Drake equations, but it's a wrong impression due to a poor "intuitive"
understanding of what statistics is! In fact, neither the mode nor the median are the
"really important" values: the really important value for N is the MEAN VALUE! Now if
you look at the thin curve in Figure 2 below (i.e. the lognormal distribution arising from
the Central Limit Theorem), you see that this curve has a LONG TAIL ON THE RIGHT! In
other words, it does NOT immediately go down to nearly zero beyond the peak of the
mode. Thus, when you actually compute the mean value, you should not be too
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