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

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a 2
5,,.2
a']
Our next goal is to find the cumulants of the
e- P[ e 2 - 3 e 18 + 2 e 6
ET _Di stance. 1n principle, we could compute all
the cumulants K; from the generic i-th moment
µ; by virtue of the recursion formula (see ref. [8])
'
i- 1 ( i -1)
'
(126)
k=i
k - 1
K; = JI; -z:
Kk J.111-k ·
ln practice, however, here we shall confine
ourselves to the computation of the fust four
cumulants because they only are required to find
the skewness and kurtosis of the distribution ( 113).
Then, the first four cumulants in terms of the first
four moments read:
These equations yield, respectively:
_!!_
a 2
K .=Ce3el8_
(128)
(129)
(130)
Sa '
5a2
4a2
a 2
2a2 ]2
C3 [ e- 9- -4e9 -3e9
+12 e3 -6e_9_
... (132)
and the kurtosis
4 0"2
2 a 2
K
­
(K 4)2 =e 9 +2e 3 +3e 9 -6 .
(133)
2
Next we want to find the mode of this
distribution, i.e. the abscissa of its peak. To do so,
we must fust compute the derivative of the
probability density function fET_Oistame(r) of (113),
and then set it equal to zero. This derivative is
actually the derivative of the ratio of two functions
of r, as its plainly appears from (113). Thus, let us
set for a moment
(134)
where
"E"
stands
for
"exponent,"
Upon
differentiating,
one gets
(13 1)
= - 1 •[ln[£.] - µ)·(- 3).!..
(135)
c,2
,-3
,.
2
4a2
4p [
sa
5a2
a'
2a']
=C4 e-3
e- 9- -4 e- 9- - 3e_9_ + 12 e3 - 6e_9_
But the probability density function (113) now
reads
3
e-E(r)
From these we derive the skewness
fET Distana:(r)= ~ ·--
(136)
-
..,;2,rc,
r
So that its derivative is
dfET_Disiana:, (r)
3
-e- E(r) E' (r)· r-1 ·e- E(r)
dr
= &a.
,.2
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