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/ ,'Pett err1e1At 652 one,
We now want to study this probability
distribution in detail. Our next questions are:
I) What is its mean value?
2) What arc it\ vanancc and standard
deviation?
3) What are its moment~ to any higher order?
4) What are its cumulants?
5) What are its skewnes~ and kurtosis?
6) What are the coordinate~ of it~ peak, 1.e.
the mode (peak ahsci\sa) and its ordinate'?
7) What is its median?
The first three points in the list are all covered
by the following theorem: all the moments of ( 113)
are given by (here k is the generic and non-
negative integer exponent, i.e. k = 0, I, 2,3, ... ~ 0)
l, ' 3
= r e
o r &a
-k ,u ! ' CT
=C"e ·'e IS ( 115)
To prove this result, one first transforms the above
integral by virtue of the substitution
( 116)
Then the new integral in z i~ then seen to reduce to
the known Gau~sian integral (53) and, after \everal
reductions that we skip for the ~ake of brevity.
( 115) follows from (53). In other words. we have
proven that
(ET_Distancc 1 ) = C1 e ( 117)
Upon \e!!ing k = {I into ( 117), the
normalization condition for fET_Di,i,m"'(r) follow~
( 118)
48
Upon setting k = I into (117). the important
mean value of the random variable ET _Distance
is.fOund
(bT_Distance) = Cc .i c rn ( 119)
Upon setting k = 2 into ( 117). the mean value of
the square of the random variable ET_Distance is
found
( 120)
The variance of ET_Distance now follow~ from
the las! two formulae with a few reductiom:
a~T_D,mn"' = ( Ef_Distance 2 )-(Ef_Distance) 2
( 121 l
So, the variance of ET_Distance is
( 122)
The ~4uare root of this is the important
standard deviation of the ET_Distance random
variable
_S c'fe'
_ , .1 , 18 , 9 _
O"ET_D1'!an<"-C{ l l 1. ( 123)
The third moment is obtained upon setting
k =3 into (117)
"
(Er_Distance 1 ) = C 1 e-p e 2. ( 124)
Finally, upon ~ctting k = 4 into ( I 17). the fourth
moment of ET Di\tancc i~ found
' '
Ef_Distance = C e :i e 9
( ') 0 - p C ( 125)
UNCLASSIFIED/ }FQA: QFFlfiil1l1k W~i Q•lklf 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.