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AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

U.S. Department of War · 2010-03-11 · 55 pages · text from the file's own layer

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 OFFIEil.t.k W&i Q~lls¥
1s := 3 · o-109 s := -s cr s := 1-109
·o 10
μfp := fpfp := 100 crfp := 100
1
ne := 1 μne := ne crne := -
,/3
fl :=~ μfl := fl crt1 = ~
100 100
fi := 20 μfi := fi crfi := ~
100 100
fc := 20
-100 μfc := fc 10
crfc := -
100
10000
fl. :=-- μfl. := fl. 1000
crfl. := --
1010 101 0
s-fp -ne-fl -fi.fc .fl. - = 3500
Table 1. Input values (i.e. mean values and standard devi ations) for the seven Drake uni fo rm random variables D;.
The fi rst column on the left lists the seven input sheer numbers that al so become the mean values (middle co lumn).
Finally the last column on the right lists the seven input standard dev iations. The bottom line is the cl assical Drake
equation ( 1).
3.2 STEP 6: COMPUTING THE LOGS
OF THE 7 UNIFORMY
DISTRIB UTED DRAKE RANDOM
VARIABLES D;
Intuiti vely speakiJ1g, the natural log of a
unifonnly disuibuted random vari able may not be
another uniforml y distributed random variable! This
is obvious fro m the tri vial diagram of y = ln (x)
shown below:
Natural logarithm of x
-
_,,,,,,,,_,,. ~
r 2 3 4 5
POSITIVE independent variable x
Figure 1. The simple function y = ln (x).
So, if we have a unifo rml y di stributed random
variable D; with lower limit a;and upper limit b;, the
random variable
Y; =ln(D;) i = l, ...,7 (23)
must have its range limited in between the lower limit
l11 (a;) and the upper limi t ln(b,). In other words, this
are the lower and upper limits of the relevant
probability density functio n fr;(y). But what is the
actual analytic expression of such a pdf/ . To find it,
we must resort to the general transformation law for
random variables, defined by equation (9). Here we
obviously have
y =g(x) =ln(x) (24)
That, upon inversion, yields the single root
(25)
On the other hand, differentiating (24) one gets
UNCLASSIFIED//FOR OFFIEil.t.k W&i QNls¥
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Official release, from the pursue collection. The PDF is mirrored here; the original link is above. 55 pages are in the text index: search them above, or from the library's search.