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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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So, let us take the natural logs of both sides of the
Statistical Drake equation (3) and change it into a
sum:
It is now convenient to introduce eight new (positive)
random variables defined as follows:
Y=ln(N)
{ (5)
Y;=ln(D;) i=l,...,7.
Upon inversion, the first equation of (5) yields the
important equation, that will be used in the sequel
(6)
We are now ready to take STEP THREE.
STEP 3: THE TRANSFORMATION LAW
OF RANDOM VARIABLES
So far we did not mention at all the problem:
"which probability disttibution shall we attach to
each of the seven (positive) random variables D; ?"
It is not easy to answer this question because we
do not have the least scientific clue to what
probability distributions fit at best to each of the
seven points listed in Section 1.
Yet, at least one trivial error must be avoided:
claiming that each of those seven random variables
must have a Gaussian (i.e. normal) distribution. In
fact, the Gaussian distribution, having the well
known bell-shaped probability density function
(a ;,: o) (7)
has its independent variable y ranging between --oo
and oo and so it can apply to a real random variable
Y only, and never to positive random variables like
those in the statistical Drake equation (3). Period.
Searching again for probability density functions
that represent positive random variables, an obvious
choice would be the gamma distributions (see, for
instance, ref. [6]). However, we discarded this choice
too because of a different reason: please keep in mind
that, according to (5), once we selected a particular
type of probability density function (pdt) for the last
seven of equations (5), then we must compute the
(new and different) pdf of the logs of such random
variables. And the pdf of these logs certainly is not
gamma-type any more.
It is high time now to remind the reader of a
certain theorem that is proved in probability courses,
but, unfortunately, does not seem to have a specific
name. It is the transformation law (so we shall call
it, see, for instance, ref. [5]) allowing us to compute
the pdf of a certain new random variable Y that is a
known function Y = g(X) of another random
variable X having a known pdf. In other words, if the
pdf fx (x) of a certain random variable X is known,
then the pdf fr(Y) of the new random variable Y,
related to X by the functional relationship
y = g(X) (8)
can be calculated according to this rule:
1) First invert the corresponding non-probabilistic
equation y = g(x) and denote by X; (y) the
various real roots resulting from the this
inversion.
2) Second, take notice whether these real roots may
be either finitely- or infinitely-many, according
to the nature of the function y = g(x).
3) Third, the probability density function of Y is
then given by the (finite or infinite) sum
(9)
where the summation extends to all roots x;(Y) and
lg' (x; (y)~ is the absolute value of the first
derivative of g(x) where the i-th root x;(Y) has
been replaced instead of x.
Since we must use this transformation law to transfer
from the D; to the Y; =ln(D;), it is clear that we
need to start from a D; pdf that is as simple as
possible. The gamma pdf is not responding to this
need because the analytic expression of the
transformed pdf is very complicated (or, at least, it
looked so to this author in the first instance). Also,
the gamma distribution has two free parameters in it,
and this "complicates" its application to the various
meanings of the Drake equation. Tn conclusion, we
discarded the gamma distributions and confined
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32 Not linked to a story yet.
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