Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

UNCLASSIFIED/ /1"91t 91"1"1!111it 1!191!! 9HLY
fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR
SETI, ina.mmch as the nearest ET Civilization might lie atjmt 1 sigma= 2670-1309 = 1361 light years
from us.
From Figure 6, we see that the probability of
finding Ex!raTeITestrials i\ pra<.:tically Lero up lo a
distance of about 500 light years from Earth. Then
it qans increa~ing with the increasing distance
from Earth, and reaches its maximum at
_!'_ ,r
r,,rn,c=r1,mJ.=Ce •1 e <) :=c]9JJ]igh1ycar.;. (152)
This is the MOST LIKELY VALUE of the
distance ut which we can expect to find the
nearest ExtraTerrestrial civilizatio11.
It is not, however. the mean value of the
probability distribution (113) for .f1.,:_,Ji,,.u,1e(r), In
fact, the probability density ( 113) has an infinite
tail on !he right, as clearly shown in Figure 6, and
hence ii\ mean value mus! be higher than its peak
value, As given by (119). its mean value is
p "
r,,,,,""-''"''".=Ce ie 1x ~2670 light years. (153)
This is the MEAN (value of the) DISTANCE
at which we can expect to find ExtraTerrestrials.
After having found the above two distances ( 1933
and 2671) light years, re~pectively), the next natural
4uestion that ari~es is: '·what is the range, forth am!
back around the mean value or the di~tance. within
which we can expect to find ExtraTerreqrials with
"the highest hopes?," The answer to this question
is given by the notion of standard deviation. that
we already found to he given by (123)
... ( 154)
More precisely, this i\ the \O called I-sigma
(distance) level. Probability theory then shows that
the nearest ExtraTerrestrial civilization is expected
to he located within this range, i.e. within the two
distances of (2671)-1Jl)9) = 1361 light years and
(2670+1309) = 3979 light year\, with probability
53
given hy the integral of fET_Dostan<e(r) taken in
between these two lower and upper limits, that is:
r.'97911 eIll a"""
J, - • .hTD1,rnn,o(r)dr:=c0.75=75% (155)
13/Jlilglllyea"
In plain words: with 75% probability. the nearest
ExtraTerrestrial civilization is located in between
the distances of 1361 and 3979 light years from us,
having assumed the input values to the Drake
Equation given by Table I. If we change those
input values, then all the numbers change again.
9. THE "DATA E~RICH.\1E~T
PRI~CIPLE" AS THE BEST CLT
CONSEQUENCE UPON THE
STATISTICAL DRAKE EQUATION
(ANY NUMBER OF FACTORS
ALLOWED)
As a fitting climax to all the statistical
equations developed so far, let us nmv state our
"DATA ElVRICHMENT PRINCIPLE," It simply stutes thut
"The llif.:her the IVumber of Factors i11 the
Statistical Drake equation, The Better,"
Put in this simple way, it simply looks like a
new way of saying that the CLT lets the random
variable Y approach the normal distribution \Vhen
the number of terms in the ~um (4) approaches
infinity. Ami this i\ !he ca\e, indeed. However, our
"Data Enrichment Principle" has more profound
methodological consequences that we cannot
explain now, but hope to describe more precisely
m one or more commg papers.
CONCLUSIONS
We have sought to extend the classical Drake
equation to let it encompass Statistics and
Probability.
This approach appears tn pave the way to
future, more profound investigations intended not
only to associate ·'error bars" to each factor in the
Drake equation, but especially to increase the
number of factors themselves. In fact, this seems to
be the only way to incorporate into the Drake
UNCLASSIFIED/ ;rett err1e1xc USE OHEI