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

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(82)
2 fx '
erf(x)= r
e-:-dz
(85)
This is "how likely" the most likely number of
ExtraTerrestrial Civilizations in the Galaxy is, i.e.
it is the peak height in the lognormal probability
density function f N(n) .
Next to the mode, the median m (ref. (9)) is one
more statistical number used to characterize any
probability
distribution.
It
is defined
as
the
independent
variable abscissa
m such
that a
realization of the random variable will take up a
value lower than m with 50% probability or a value
higher than m with 50% probability again. In other
words, the median m splits up our probability
density in exactly two equally probable parts. Since
the probability of occurrence of the random event
equals the area under its density curve (i.e. the
definite integral under its density curve) then the
median m (of the lognormal distribution, in this
case) is defined as the integral upper limit m:
(111(11 )-,,1)2
f
111 JN(n)dn = f"'_!__ rd-
e---;;;;r- ==
(83)
Jo
Jo n v 2na
2
In order to find m , we may not differentiate (83) with
respect to m , since the "precise" factor ½ on the
right would then disappear into a zero. On the
contrary, we may try to perform the obvious
substitution
zzO
(84)
into the integral (83) to reduce it to the following
integral defining the error function erf(z)
-,/ Ji 0
Then, after a few reductions that we skip for the sake
of brevity, the full equation (83) is turned into
(86)
that is
e,f[ln(m) - p)
== 0
(87)
✓2a
Since from the definition (85) one obviously has
erf(0)=0, (87) becomes
ln(m)- µ == 0
(88)
✓2a
whence finally
Irredian = m =e" I.
(89)
This is the median of the lognormal distribution of
N.
In other words, this is the number of
ExtraTerrestrial civilizations in the Galaxy such
that, with 50% probability the actual value of N will
be lower than this median, and with 50% probability
it will be higher.
ln conclusion, we feel useful to summarize all the
equations that we derived about the random variable
N in the following Table 2.
Random variable
Probability distribution
Probability density function
N =number of communicating ET civilizations in Galaxy
Lognormal
(111(11}--p)'
1
1
---;;;;r­
fN(n) =­· ..n,; e
(n z O)
n
2na
Mean value
(T2
-
(N) =e" e 2
Variance
a1 =e2" ea2 (ea2 -1)
Standard deviation
a'
a N =e" e2 .Jea' -1
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