Documents / Report
This Defense Intelligence Agency reference document, dated 11 March 2010, is one of the advanced technology reports produced in FY 2009 under the Advanced Aerospace Weapon System Applications (AAWSA) program. It introduces the Statistical Drake Equation, which replaces each factor of Frank Drake's 1961 equation with a uniform random variable to estimate how far away the nearest extraterrestrial civilization is. In the worked example, there is a 75% probability that the nearest civilization lies between 1,361 and 3,979 light years from Earth.
UNCLASSIFIED/ /1"91t 91"1"1!111it 1!191!! 9HLY
Our next goal is to find the cumulants of the
ET _Distance. In principle, \Ve could compute all
the cumulants K, from the generic i-th moment
J.1, by virtue of the recursion fommla (see ref. [8])
(126)
In practice, however, here we ~hall confine
ourselves to the computation of the first four
cumulants because they only arc required to find
the skewness and ku11mis of the distribution (113).
Then, the first four cumulants in terms of the first
four moments read:
These equations yield, respectively:
_!'_ ~
K1 =Ce-' e 1~. (128)
(129)
[
c' .'i ,-,'
+,, ~, lK,=C·1 e-·"e 2 - 3 e IH ( 130)
'
K_., = (131 l
>;, [ <c' :,('l"' _.('l", a'
"a' l=C 4 e ' ' ' -4, " - 3e " + 12 e .1 -6e 9
From thc~c we derive the ~kcwncss
49
r
a- :,('l"'
+,,: l,, 11 e -:, - J e IS
\
... (132)
and the ku1tosis
K _.(l", "- z(l"'
--'-=e 9 +2e-1 +3e 9 -6. (133)
(K, )'
Next we want to find the mode of this
distribution, i.e. the abscissa of its peak. To do so,
we must first compute the derivative of the
probability density function f~.i_n,_,,,""'(r) of ( 113),
and then set it equal lo zero. This derivative i\
actually the derivative of the ratio of two functiom
of r, as its plainly appears from (113). Thus, let us
set for a moment
where "E"
differentiating,
one gets
stands fm
(134)
"exponent," Upon
£(,Jc,:, ,[,{~:]-p]c' c' (-3),'
,'
(135)
But the probability density function (113) now
reads
3 e -f.(, l
/1 I D1,lall<C(r)= ~
...J2na- r
(136)
So that its derivative is
<'.(/ETDi,ian"''(r) 3 -e E(l')E'(r)•r-l·e £(/')
dr Ea. ,.2
UNCLASSIFIED/ /P81t 8PPll!llltt l!l!ili 8111!'/ Not linked to a story yet.
Report, from the dia 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.