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,¥
(36)
Thus, the characteristic function of the natural log
ofthe Drake uniform random variable D; is given by
3.3 STEP 7: FINDING THE
PROBABILITY DENSITY
FUNCTION OF N, BUT ONLY
NUMERICALLY NOT
ANALYTICALLY
Having found the characteristic functions
y/s) of the logs of the seven input random
variables D; , we can now immediately find the
characteristic fu □ ctio □ of the random variable Y =
ln(N) defined by (5). 1n fact, by virtue of (4), of the
well-known Fourier transform property stating that
"the Fourier transform of a convolution is the product
of the Fourier transforms", and of (37), it
immediately follows that y(() equals the product
of the seven Cf> y1 (s):
The next step is to invert this Fourier transform in
order to get the probability density function of the
random variable Y = ln(N). In other words, we must
compute the following inverse Fourier transform
-oo
oo [ 7 bl +)( i+ j( l= - 1 fe -i(y IT ;_ - a;_ dc;'.(39)
2n - co i= I (b; a;)(l+ Js)
This author regrets that he was unable to compute the
last integral analytically. He had to compute it
numerically for the particular values of the 14 a; and
b; that follow from Table 1 and equations 17. The
result was the probability density function for Y =
ln(N) plotted in the following Figure 2.
...>- PROB. DENSITY FUNCTION OF Y=ln(N)
0 0.4
C
0
·.:::,
..2
Q 0.3
-f 0.2
ii
"'~ 0.1
:z;
..8
•
~I....-, / 'I \\O O I 2 3 4 5 6 7 8 9 10 11 12ell,
Independent variable Y = ln(N)
Figure 2. Probability density function of Y = ln(N)
computed numericaJly by virtue of the integral (39).
The two "funny gaps" in the curve are due to the
numeric limitations in the MathCad numeric solver
that the author used for this numeric computation.
We are now just one more step from finding the
probability density of N, the number of
ExtraTerrestrial Civilizations in the Galaxy predicted
by our Statistical Drake equation (3) . The point here
is to transfer from the probability density function of
Y to that of N, knowing that Y = ln(N) , or
alternatively, that N=exp(Y), as stated by (6). We
must thus resort to the transformation law of random
variables (9) by setting
(40)
This, upon inversion, yields the single root
(41)
On the other hand, differentiating (40) one gets
(42)
where (41) was already used in the last step. The
general transformation law (9) finally yields
UNCLASSIFIED/ /EAR AFFJCJA 1 1!SF QN 1Y
37