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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8. Finding the Probability Distribution of the Et-Distance By
Virtue of the Statistical Drake Equation
Having solved the statistical Drake equation by finding the lognormal distribution, we
are now in a position to solve the ET-DISTANCE problem by resorting to statistics again,
rather than just to the purely deterministic Distance Law (5), as we did in Section 2.
This is "scientifically more serious" than just the purely deterministic Distance Law (5)
inasmuch as the new statistical Distance Law will yield a PROBABILITY DENSITY for the
Distance, with the relevant mean value and standard deviation . In other words, the
Distance Law (5) itself becomes a random variable whose probability distribution, mean
value and standard deviation must be computed by "replacing" into (5) the fact that N
is now known to follow the lognormal distribution . This is mathematically described in
detail in Section 7 of Appendix A.
The important new result is the PROBABILITY DENSITY FOR THE DISTANCE, the
equation of which is
(9)
holding for r ;,: O. This is equation (114) of Appendix B.
Starting from this equation, the MEAN VALUE OF THE random variable ET_DISTANCE is
computed as
J.I cr2
(Er_Distance) =Ce3 e""jg (10)
which is equation (119) of Appendix B, and finally the ET_DISTANCE STANDARD
DEVIATION
_!!. a2Kr2
- 3 18 9
CJ ET_Distanu: - C e e e - l (11)
which is equation ( 123) of Appendix B. Of course, all other descriptive statistical
quantities, such as moments, cumulants etc. can be computed upon starting from the
probability density (9), and the resu lt is Table two hereafter, that is Table 3 of Appendix
B.
Finally, to complete this section, as well as this "introduction to the statistical Drake
equation," the numerical values that equations (10) and (11) yield for the Input Table 1
are determined. They are, respectively:
_}!_ CTl
r,11ea11 _ m l ue = Ce 3 e 18 ;::: 2,670 light years (12)
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