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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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UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥
where (25) was already used in the last step. By
virtue of the uniform probability density function
(I 0) and of (26), the general transformation law (9)
finally yields
1n other words, the requested pdf of Y; is
eY ,--,---,,-----,-.,..,
fr,(y) =-- i = l, ...,7 lm(a;) :,;y:,;in(b;)I (28)
b; - a;
Probability density functions of the natural logs of
all the uniformly distributed Drake random
variables D; .
This is indeed a positive function of y over the
interval !n(a;):,; y:,; ln(b;) , as for every pdf, and it is
easy to see that its normalization condition is
fulfilled:
... (29)
Next we want to find the mean value and
standard deviation of Y; , since these play a crucial
role for future developments. The mean value (Y;) is
given by
!n(b;) ( ) s.•n(b;) y,e Y
(Y.) = J. Y· fr y dy = -- dy1 Jn(a,) ' !n(a,)b; -a;
b; [in(b; )-1]- a,-[in(a; )-1]= --'-''----''--'--'--=--'-"-"--'-'---= (30)
This is thus the mean value of the natural log of all
the uniformly distributed Drake random variables
D;
In order to find the variance also, we must first
compute the mean value of the square of Y;, that is
( 2) J.in(b.) 2 { .) J.!n(/J.) y2 •e Y
Y; = y ·fry dy= --dy
h1(<1;) ' !n(a;) b; - Cl;
= b; ~ 2 (b; )- 21n(b; )+ 2] - a; [ln 2 (a; )- 2ln(a; )+ 2)
b; - a;
... (32)
The variance of Y; = ln(D;) is now given by (32)
minus the square of (31) , that, after a few reductions,
yield:
2 2 a .b.[ln(b .)- ln(a .)]2
a - a _ 1_ , , , , (33)r, - In(o,) - (b; _ a;)2
Whence the corresponding standard deviation
1 _ a;b; [in(b;)- in(a;)]2
(34)
(b; - a;)2
Let us now turn to another topic: the use of
Fourier transforms, that, in probability theory, are
called "characteristic functions," Following again the
notations of Papouli s (ref. [5]) we call "characteristic
function", <Dy, (s) , of an assigned probability
distribution Y; , the Fourier transform of the relevant
probability density function, that is (with j = ~)
The use of characteristic functions simplifies things
greatly. For instance, the calculation of all moments
of a known pdf becomes trivial if the relevant
characteristic function is known, and greatly
simplified also are the proofs of important theorems
of statistics, like the Central Limit Theorem that we
will use in Section 4. Another important result is that
the characteristic function of the sum of a finite
number of independent random variables is simply
given by the product of the corresponding
characteristic functions. This is just the case we are
facing in the Statistical Drake equation (3) and so we
are now led to find the characteristic function of the
random variable Y; , i.e.
UNCLASSIFIED/,'FOA OFlilCI0 I. P!SE ON! X
36 Not linked to a story yet.
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