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,¥
(82) 2 fx 'erf(x)= r e-:-dz (85)
This is "how likely" the most likely number of
ExtraTerrestrial Civilizations in the Galaxy is, i.e.
it is the peak height in the lognormal probability
density function f N(n) .
Next to the mode, the median m (ref. (9)) is one
more stati stical number used to characterize any
probability distribution. It is defined as the
independent variable abscissa m such that a
realization of the random variable will take up a
value lower than m with 50% probability or a value
higher than m with 50% probability again. In other
words, the median m splits up our probability
density in exactly two equally probable parts. Since
the probability of occurrence of the random event
equals the area under its density curve (i.e. the
definite integral under its density curve) then the
median m (of the lognormal distribution, in this
case) is defined as the integral upper limit m:
(111(11 )-,,1)2
f111 JN(n)dn = f "'_!__ rd- e---;;;;r- == (83)
Jo Jo n v 2na 2
In order to find m , we may not differen tiate (83) with
respect to m , since the "precise" factor ½ on the
right would then disappear into a zero. On the
contrary, we may try to perform the obvious
substitution
zzO (84)
into the integral (83) to reduce it to the following
integral defining the error function erf(z)
-,/ Ji 0
Then, after a few reductions that we skip for the sake
of brevity, the full equation (83) is turned into
(86)
that is
e,f[ln(m) - p)== 0 (87)
✓2a
Since from the definition (85) one obviously has
erf(0)=0, (87) becomes
ln(m)- μ == 0 (88)
✓2a
whence finally
Irredian = m =e" I. (89)
This is the median of the lognormal distribution of
N. In other words, this is the number of
ExtraTerrestrial civilizations in the Galaxy such
that, with 50% probability the actual value of N will
be lower than this median, and with 50% probability
it will be higher.
ln conclusion, we feel useful to summarize all the
equations that we derived about the random variable
N in the fo llowing Table 2.
Random variable
Probability distribution
Probability density function
N =number of communicating ET civilizations in Galaxy
Lognormal
(111(11}--p)'
1 1 ---;;;;r-
fN(n) =- · ..n,; e (n z O)
n 2na
Mean value
(T2
-
(N) =e" e 2
Variance a 1 =e2" ea2 (ea2 -1)
Standard deviation
a'
a N =e" e2 .Jea' -1
UNCLASSIFIED/ /fOR OFfl&IAk Wlii QNls¥
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