AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

This Defense Intelligence Reference Document, dated 11 March 2010, was prepared by the Defense Intelligence Agency's Defense Warning Office under the Advanced Aerospace Weapon System Applications Program. It introduces the Statistical Drake Equation, which treats each Drake factor as a random variable with a mean value and a standard deviation. Using its example inputs, the paper estimates that the nearest extraterrestrial civilization lies between 1,361 and 3,979 light years away with 75% probability. The author's 2008 International Astronautical Congress paper is attached as an appendix.

From the source:Release of 2026-09-18 Incident: 3/11/10, Las Vegas, Nevada. Released with redactions. This document is a Defense Intelligence Reference Document (DIRD), a technical reference format used by the Defense Intelligence Agency (DIA) to capture baseline knowledge on a specific topic for later analytic use. DIRDs are best understood as reference and synthesis products rather than as original research. It is one of 38 DIRDs produced under the Advanced Aerospace Weapon System Applications Program (AAWSAP) between 2009 and 2011. Because AAWSAP’s scope permitted a broad range of supporting topics, not every DIRD in the series directly concerns aerospace systems or future threat assessment. The following summary reflects the DIRD’s scope and framing at the time of writing and should not be read as implying current validation of the concepts discussed. This DIRD introduces the Drake Equation, a well-known thought framework for estimating how many communicative extraterrestrial civilizations might exist in the galaxy. It reformulates the equation in statistical terms, arguing that the usual approach of assigning fixed values to its variables is too simplistic because major inputs are uncertain and are better modeled as probability distributions. Using that approach, it concludes that, if one accepts the underlying logic of the Drake Equation, the estimated number of communicating civilizations should be treated as a range of possible values, and that the likely distance between neighboring civilizations can likewise be expressed statistically rather than as a single figure. The document is primarily a mathematical and methodological exercise, and its worked examples rely on assumed values to illustrate the framework rather than to establish a firm astrophysical estimate. Overall, it is an attempt to formalize uncertainty within the Drake framework rather than an attempt to bound the actual likelihood, prevalence, or proximity of extraterrestrial civilizations.

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Let us go back to equation (104). Since N is
now a random variable (obeying the lognorma1
distribution), it fo ll ows that the ET_Distance must
be a random variable as well. Hence it must have
some unknown probability density function that
we denote by
(106)f ET_Di stana, (r)
where r is the new independent variable of such a
probability distribution (it is denoted by r to
remind the reader that it expresses the three
dimensional radial distance separating us from the
nearest ET civilization in a full spherical symmetry
of the space around us).
The question then is: what is the unknown
probability distribution (106) of the ET_Distance?
We can answer this question upon making the two
formal substitutions
N ➔ x
{ (107)
Er_distance ➔ y
into the transformation law (8) for random
variables. As a consequence, ( I04) takes form
I
C -
y = g(x) = -,- = C • x 3 . (108)
v;;
In order to find the unknown probability density
f ET_Di stru,., (r) , we now to apply the rule (9) to
(108). First, notice that (108), when inverted to
yield the various roots x; (y), yields a single real
root only
(109)
Then, the summation in (9) reduces to one term
only.
Second, differentiating (108) one finds
4
C --
g ' (x) =-- ·x 3 . (110)
3
Thus, the relevant absolute value reads
(111)
Upon replacing (111) into (9), we then find
This is the denominator of (9) . The numerator
simply is the lognormal probability density
function (56) where the old independent variable x
must now be re-written in terms of the new
independent variable y by virtue of (109). By
doing so, we finally an-ive at the new probability
density function f y (y)
Rearranging and replacing y by r, the final form
is:
2
(In[~]-11]
3 1 - 2u'
f ET_d istana, (r) = - • ~ •e (113)2r 'I/ L.Jr a-
Now, just replace C in ( 113) by virtue of ( 105).
Then:
We have discovered the probability density
function yielding the probability of finding the
nearest ExtraTerrestrial Civilization in the
Galaxy in the spherical shell between the
distances r and r+dr from Earth:
(,{ 6 Ri.t11a~~ hc0 1a.Q}l'r
3 l 2 0- l
fET_Di stanaJr)=-· ~ ·e2r "1.,1r a
(114)
holding for r .:: 0 .
STATISTICAL PROPERTIES OF THIS
DISTRIBUTION
UNCLASSIFIED/ /FOR &FFIGIAk lalii ODIL:¥
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