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.

UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥
We now want to study this probability
distribution in detail. Our next questions are:
l) What is its mean value?
2) What are its variance and standard
deviation?
3) What are its moments to any higher order?
4) What are its cumulants?
5) What are its skewness and kurtosis?
6) What are the coordinates of its peak, i.e.
the mode (peak abscissa) and its ordinate?
7) What is its median?
The first three points in the list are all covered
by the following theorem: all the moments of ( 113)
are given by (here k is the generic and non
negative integer exponent, i.e. k = 0, I, 2, 3,... ~ 0)
(Er_Distance*) = f'/ ·f ET_Distana,(r)dr
( 1n[~ ]-μr
dr=fr*-r~(T -e
(115)
To prove this result, one first transforms the above
integral by virtue of the substitution
(116)
Then the new integral in z is then seen to reduce to
the known Gaussian integral (53) and, after several
reductions that we skip for the sake of brevity,
(115) follows from (53). In other words, we have
proven that
μ 2 (J2
(Er_Distancek) = Ck e - k3 / •18 . (117)
Upon setting k = 0 into (117), the
normalization condition for f ET_Distan"' (r) follows
if ET Distame(r)dr = l . (118)
0 -
Upon setting k = l into (117), the important
mean val11e of the random variable ET_Distance
isfo1111d
f.-l a z
(Ef_Distance) =C e3 e18 . (119)
Upon setting k = 2 into (l 17), the mean value of
the square of the random variable ET_Distance is
found
2 2 2
- p - (1
(Ef_Distance 2 ) = C2 e 3 e 9 (120)
The variance of ET_Distance now follows from
the last two formulae with a few reductions:
CT~T_Distan"' = (Ef_Distance2 )- (Ef_Distance)2
(121)
So, the variance ofET_Distance is
(122)
The square root of this is the important
standard deviation of the ET_Distance random
variable
_f!_ (11 ,-;i--
(TET_Di S(ana, =C e 3 e• 8 ~e9 -1 . (123)
The third moment is obtained upon setting
k = 3 into ( 117)
(1 2
(Er_Distance 3 ) = C 3 e-p e2 (124)
Finally, upon setting k = 4 into ( 117), the fourth
moment of ET_Distance is found
4 8 l
(Er_Distance 4 )=C4 e3 -μ e9(1 (125)
UNCLASSIFIED//POR 9Pfl@IAL H§E 8PUs:¥
48