AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

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We now want to study this probability
distribution in detail. Our next questions are:
l) What is its mean value?
2) What are its
variance and standard
deviation?
3) What are its moments to any higher order?
4) What are its cumulants?
5) What are its skewness and kurtosis?
6) What are the coordinates of its peak, i.e.
the mode (peak abscissa) 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)
(Er_Distance*) = f' / ·f ET_Distana,(r)dr
(1n[ ~ ]-µr
dr
=fr*-r~(T -e
(115)
To prove this result, one first transforms the above
integral by virtue of the substitution
(116)
Then the new integral in z is then seen to reduce to
the known Gaussian integral (53) and, after several
reductions that we skip for the sake of brevity,
(115) follows from (53). In other words, we have
proven that
µ
2 (J2
(Er_Distancek) = Ck e - k3 / •18 .
(117)
Upon
setting
k = 0
into
(117),
the
normalization condition for f ET_Distan"' (r) follows
if ET Distame(r)dr = l .
(118)
0
­
Upon setting k = l into (117), the important
mean val11e of the random variable ET_Distance
isfo1111d
f.-l
a z
(Ef_Distance) =Ce3 e18 .
(119)
Upon setting k = 2 into (l 17), the mean value of
the square of the random variable ET _Distance is
found
2
2
2
- p
-
(1
(Ef_Distance2 ) = C2 e 3 e 9
(120)
The variance of ET_Distance now follows from
the last two formulae with a few reductions:
CT~T_Distan"' = (Ef_Distance2 )-(Ef_Distance)2
(121)
So, the variance ofET_Distance is
(122)
The square root of this is the important
standard deviation of the ET_Distance random
variable
_f!_
(11 ,-;i-­
(TET_Di S(ana, =C e 3 e• 8 ~e9 -1 .
(123)
The third moment is obtained upon setting
k = 3 into ( 117)
(12
(Er_Distance3 ) = C3 e-p e2
(124)
Finally, upon setting k =4 into ( 117), the fourth
moment of ET _Distance is found
4
8 l
(Er_Distance4)=C4 e3 -µ e9(1
(125)
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