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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/ /1"91t 91"1"1!111it tl!IL 9HL I 8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical Drake Equation Having solved the statistical Drake equation by finding the lognormal distribution, we are now in a position to solve the ET-DISTANCE problem by resorting to statistics again, rather than just to the purely deterministic Distance Law (5), as we did in Section 2. This is "scientifically more serious" than just the purely deterministic Distance Law (5) inasmuch as the new statistical Distance Law will yield a PROBABILITY DENSITY for the Distance, with the relevant mean value and standard deviation. In other words, the Distance Law (5) itself becomes a random variable whose probability distribution, mean value and standard deviation must be computed by "replacing" into (5) the fact that N is now known to follow the lognormal distribution. This is mathematically described in detail in Section 7 of Appendix A. The important new result is the PROBABILITY DENSITY FOR THE DISTANCE, the equation of which is rhI oR,\,./,,.,. J, r fET_D1atan..,(r) =~-,Ji; CT· e 2u 2 holding for r2'.0. This is equation (114) of Appendix B. Starting from this equation, the MEAN VALUE OF THE random variable ET_DISTANCE is computed as ( 9) _i!_ er (ET_Distance)=Ce 3 e 18 (10) which is equation (119) of Appendix B, and finally the ET_DISTANCE STANDARD DEVIATION -• "--~- '.' IS 'J al'."I_Di,ta,rn, - Ce e e - ! (11) which is equation (123) of Appendix B. Of course, all other descriptive statistical quantities, such as moments, cumulants etc. can be computed upon starting from the probability density (9), and the result is Table two hereafter, that is Table 3 of Appendix B. Finally, to complete this section, as well as this "introduction to the statistical Drake equation," the numerical values that equations (10) and (11) yield for the Input Table 1 are determined. They are, respectively: - r,"""" ,·"iuc = Ce 3 e 18 ::e 2.670 light years (12) 18 UNCLASSIFIED/ /ffUl 8ffl@IA1s Wfii &Ullslf
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