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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.
UNCLASSIFIED/ ;CFQA: QFFl&l11J.k W&liii Q•lklf
3 -c-i,(,)[E'(r)·r+iJ.
ilia· r 2 ( 137)
Swing thi~ derivative cqual to zern means setting
£(,) ,+lcO (138)
That is, upon replacing ( 135) into ( 138), we get
Rearranging, this becomes
that is
whence
and finally
'I [c'] '--~ n -, +3μ+a· =0
,·
I [cl-" a'n - --+-
r 3 9
// ,,-
,;,wlc '= rp,·ak = CC :; C 9
( 139)
(140)
( 141)
(142)
(143)
This is the most likely ET_Distancc/Yom Earth.
How likely?
To find the value of the probability density
function h.i_Disim,., ( r) corresponding to this value
of the mode, we mus! obviously replace () into ().
After a few rearrangements, which we skip for the
sake of brevity. one gets
Peak Value of / 1. 1:_1ii,1.u,1e(r) = f1,1_1Ji,1a""'(r111,d~)
3 ,ii (T
- . (' _< . (' 18 .
c5a
so
. .. (144)
This is the peak height in the pdf _f bT IJ,"an"' (r).
Ncxt to the rnodc, the rncdian m (rcf. [91) is onc
more stati~tical numhcr used to charactcri1:c any
prnhahility di~trihutinn. It 1s defined as the
independent variable abscissa 111 such that a
rcalinttinn nf thc random variablc will takc up a
valuc lower than m with 50°/c probability or a value
highcr than 111 with 50'7c prnbability again. In nthcr
words. the median 111 sphts up our probability
dcnsity in exactly two cqually prnbablc parts. Since
the prnbability uf nccurrcm.:c nf thc randum cvcnt
cquab the area undcr its dcnsity curve (i.c. thc
definite integral under its density curve) then the
median m (of thc lngnnrmal distribution. in thi~
case) is defined as the rntegral upper limit m:
(]45)
Upnn replacing (113), this becomes
2 ( 146)
In nrdcr to find111, wc may 110! diffcrcntiate (146)
with respect to m, since the ··precise" factor './2 on the
right would thcn disappcar intu a Lcrn. On the
cnntrary, we may try tu perform thc obvinus
substitutinn
(147)
into the intcgral ( 146) to rcducc it to thc following
intcgral (85) dcfining the error functinn erf(;). Then.
aftcr a few rcductinns that we leavc In the rcadcr as
an cxcrci~c, the full equation ( 145). dcfining the
median, is turned into thc corresponding cquatinn
involving the error function er/(xl as defined hy (85):
UNCLASSIFIED/ /1"911. 91"1"!!!11tt 1!181!! 9HLY 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.