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,¥
Table 2. Summary of the Properties of the Probability Distribution That Applies
to the Random Variable ET_Distance Yielding the (average} Distance Between
Any Two Neighboring Communicating Civilizations in the Galaxy
Random variable ET_Distance between any two neighboring
ET civilizations in galaxy assuming they are
UNIFORMLY distributed throughout the
whole qalaxy volume.
Probability distribution Unnamed
Probability density function _(•{ 6 R[ata '.I l,Gola')]-μ J1
3 I 2u 2
fET_Distana,(r) =-;. ·Ji; CT ·e
Numerical constant C related to the Milky
Way size C =3 6 R 8alruy h Gala.,y "" 28,845 light years
Mean value μ u2
(Er_Distance) =C e-3 e18
Variance 2 2
2 - 2 2 a-3p 9 [ a9 - lCTET_Distance - C e e e I
Standard deviation 2
_ji_ er ✓ a
- 3 18 9 _
CTET_Distance - C e e e l
All the moments, i.e. k-th moment k2,u2
- k!!.
(Er_Dis tan eek) =c k e 3 e 18
Mode (= abscissa of the log normal peak) _ji_ er
9
rnnde =rpeak = Ce 3 e
Value of the Mode Peak Peak Value of fET_Distanu:(r) =
a'!!.3 -
·e 3 . e 18
=fET_Distana,Cr,rnde) = cfi; CT
Median ( = fifty-fifty probability value for N) _ji_
~dian = m = Ce 3
Skewness ,,., 5,,.2 ,,., ]
e-μ [ e 2 - 3e 18 +2e 6
__!!i_ =3 3
(K4 )2 8a2 5a2 4a 2 2
a 2a2 ]2
C3 [ e9 -4e- 9 -3e9 +12 e3 - 6e- 9
Ku rtosis 4 a 2 a' 2 a 2
- -
K4 _ 9 +2e 3 +3e 9 -6
(K2)2 - e
Expression of μin terms of the lower (ai)
and upper (bi) limits of the Drake uniform
input random variables Di
μ= I (r;) = I b; [ln (b;)- 1]- a;[ln(a;)- 1]
i=I ;~1 b; - a;
Expression of CT 2 in terms of the lower (ai)
and upper (bi) limits of t he Drake uniform
input random variables Di
7 7 a;b; [ln (b;)- ln(a;)f
CT 2 = Io} =Ii (b; - a; )2
i= I i=I
UNCLASSIFIED/ J'FQA QFFICiIPL. !Iii 011! Y
20