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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.
“Works”7 pages
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In practice, however, here we shall confine
oun;elves to the computation of the fin;t four
cumulants only because they only are required to
find the ~kewnes~ and kurtosis of the distribution.
Then, the fir\t four cumulants in terms of the first
four momenh read:
These equations yield, rc\pectively:
(68)
(69)
,, .
"K_1 =e.1·"e 2 (70)
From these we derive the skewne!;s
K,
(72)
and the kurtosis
(73)
Finally, we want to find the mode of the
lognormal probability densi(y function. i.e. the
abscis~a of its peak. To do ~o. we must first
compute the derivative of the probability den~ity
function f..v(n) of equation (56), and then ~et it
equal to ?ero. This derivative is actually the
derivative of the ratio of two func(iom of n, as it
plainly appear~ from (57). Thu\, let u~ set for a
moment
41
(74)
where ·'E"" ~tand~ for ·'exponent," Upon
differentiating this, one gets
(75)
But the lognormal probability density function (56),
by virtue of (74), now reads
f,(n)c ~
✓ 2nu
(76)
So that its derivative is
dfET Dt'iani,, (r)
d,
-e E(n)E.(11)•11-l·c E(n)
",,-
- e -E(" 1[1d11 )· 11 +I]_
" (77)
Setting this derivative equal to zero means setting
E(n)-11+1=0 (78)
That is, upon replacing (75),
(79)
Rearranging, this becomes
ln[n]-μ+u 2 =0 (80)
and finally
(81)
Thi.~ i.~ the mo.~t likely number of ExtraTerre.~trial
Civilization.~ in the Galaxy.
How likely? To find the value of the probability
density function fN(n) corresponding to this
value of the mode. we mu~t obviou\ly replace (81)
into (56). After a few rearrangements, one then
gets
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Report, from the dia collection. The PDF is mirrored here; the original link is under it. 55 pages are in the text index: search them above, or from the library's search.