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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.
“The Advance”6 pages
UNCLASSIFIED/ /1"91t 91"1"1!111it 1!191!! 9HLY ;-;-s C 350-10 9 p;\s := Ks 0~S C I 10' 50 Ip C pip C Ip 100 IO Olp C 100 ne ,Llne C ne one C /3 50 fl C ,u.fl := fl 100 IO ofl C 100 20 fi C ,Llfi := ti 100 IO ofi C 100 20 fc ,u.fc = fc 100 IO ofc C 100 10000 IL C ,ttL ·= tl. 1010 otL 1000 C 1010 :,;- .= "'.\"s Ip ne t1 fi fr tl. -=--· = 3500 Table 1. Input values (i.e. mean values and ~tandan.l dcviati(lm) for the seven Drake uniform random variables IJ,. The first column (Ill the left list~ the seven input sheer numbers that alsn hccomc the mean values (middle cnlumn). finally the last C(llumn nn the right lists the seven input stam.lan.l dcviatinns. The bottom line is the classical Drake cquatiun ( I). 3.2 STEP 6: COMPUTING THE LOGS OF THE 7 UNIFORMY DISTRIBUTED DRAKE RANDOM VARIABLESD; Intuitively speaking, the natural log of a uniformly distributed random variable may not be another uniformly distributed random variable! This is obvious from the trivial diagram of y = ln(x) ~huwn below: Natural logarithm of x , 2 3 4 5 POSITIVE independent variable x Figure], The simpk functiun _,, = ln(xJ. 35 Su, if we have a uniformly di~tributcd random variable D, with lowcr limit a, and uppcr limit b,, the randmn variable must have it~ range limited in between thc luwcr limit fn(a,) and thc upper limit fn(b,J. In othcr word~, this arc thc luwcr and upper limit~ of the relevant probability den~ity function fl'. (.v). But what i~ the actual analytic cxprc~sion of ~uch a pdf?. To find it, wc mu~t rcsnrt tn thc gcncral tran~formation law for random variable~, dcfincd by cquation (9). Herc we ohviomly have (24) That, upon inversion, yields the single mot (25) On the other hand. differentiating (24) one gets UNCLASSIFIED/ ;<FQA: QFFiliIAk W!iili 8HLV
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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.