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,¥
In the sequel of this paper we shall denote the
independent variable of the lognormal distribution
(47) by a lower case letter n to remind the reader that
corresponding random variable N is the positive
integer number of ExtraTerrestriaJ Civilizations in
the Galaxy. In other words, n will be treated as a
positive real number in all calculations to follow
because it is a "large" number (i.e. a continuous
variable) compared to the only civilization that we
know of, i.e. ourselves. In conclusion, from 110w 011
the log11or111al probability density f11nction of N will
be written as
(1n(11}--μ)2
1 1 -~( ) (56)
n v2na
IN n =-· r;:;- e (n~ O)
Having so said, we now turn to the statistical
properties of the lognormal distribution (55), i.e. to
the statistical properties that describe the number N
of ExtraTerrestrial Civilizations in the Galaxy.
Our first goal is to prove an equation yielding all
the moments of the lognormal distribution (56), that
is, for every non-negative integer k = 0, I, 2, . . . one
has
(57)
The relevant proof starts with the definition of the k
d1 moment
(111[11]-p)'
k I I -~
= rn •~ •.Ji; a •e dn0
One then transforms the above integral by
virtue of the substitution
In[n]= z. (58)
The new integral in z is then seen to
reduce to the Gaussian integral (53)
(we skip all steps here) and (57)
follows
Upon setting k = 0 into (56), the
normalization condition for f N (n) follows
Upon setting k =1 into (56), the important
mean value of the ra11dom variable N is found
(60)
Upon setting k = 2 into (56), the mean value
of the square of the random variable N is found
/N2) - 211 2a2
\ -e e . (61)
The variance of N now follows from the last two
formulae:
(62)
The square root of this is the important standard
deviation f onnula for the N random variable
(63)
The third moment is obtained upon setting
k = 3 into (56)
(64)
Finally, upon setting k =4, the fourth moment
of N is found
(65)
Our next goal is to find the cumulants of N. In
principle, we could compute all the cumulants K;
from the generic i-th moment p; by virtue of the
recursion formula (see ref. [8])
• i- 1 (i - 1) . (66)K; =A- L k-1 K k f.l11-k·
k=i
UNCLASSIFIED/ /rOR orrl@IAL HSE O,.LY
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