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

Text

This text came with the file. It may have mistakes. When the text and the image disagree, the image is right.

UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥
All the moments, i.e. k-th moment
Mode(= abscissa of the lognormal peak)
Value of the Mode Peak
Median(= fifty-fifty probability value for N)
Skewness
Kurtosis
Expression of µ in terms of the lower (a;) and upper
(b;) limits of the Drake uniform input random
variables D;
Expression of a 2 in terms of the lower (a;) and upper
(b;) limits of the Drake uniform input random
variables D;
-
-
JI -a2
nnnde = npeak - e e
a l
I
- 11
,
f (
)
N nTIDde = ~ •e
•e ­
-,121r a
Table 2. Summary of the properties of the lognormal distribution that applies to the random variable N = number of
ET communicating civilizations in the Galaxy.
We want to complete this section about the
lognormal probability density function (56) by
finding out its numeric values for the inputs to the
Statistical Drake equation (3) listed in Table I.
According to the CLT, the mean value µ to be
inserted into the lognormal density (56) is given
(according to the second equation (48)) by the sum of
all the mean values (Y;), that is, by virtue of (31 ), by:
Upon replacing the 14 a; and b; listed in Table 1
into (90), the following 11umeric mea11 value µ is
found
1µ,,,7.462116 1
(91)
Similarly, to get the numeric variance a 2 one
must resort to the last of equations (48) and to (33):
(92)
a 2
yielding the following numeric varia11ce
to be
inserted into the lognormal pdf (56)
la2 ,,, 1.938725 1
(93)
whence the numeric standard deviation a
1 a ,,, 1.392381 t-
(94)
Upon replacing these two numeric values (84)
and (86) into the lognormal pdf (56), the latter is
perfectly determined. It is plotted in Figure 4
hereafter as the thin curve.
In other words, Figure 4 shows the lognormal
distribution for the number N of ExtraTerrestrial
Civilizations in the Galaxy derived from the Central
Limit Theorem as applied to the Drake equation
(with the input data listed i11 Table 1).
We now like to point out the most important
statistical properties of this lognormal pdf:
1) Mean Value of N. This is given by equation (60)
with µ and a given by (91) and (94), respectively:
UNCLASSIFIED/ /FOR OFFICIO! !!SF ON! X
43