Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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fN( 11 ,muc)= ~
✓ 27r u
e -.u . e 2 (82)
This is "hhw likely" the mhst likely number of
ExtraTerrestrial Civilizatiom in the Galaxy is, i.e.
it is the peak height ill the fognonnal probability
density Junction fv (11).
Next to the mode. the median m (ref. [9]) is one
more statistical number u~ed to characterize any
probability distribution. It 1s defined a~ the
independent variable ah~cissa m ~uch that a
rcalizati(]n of the random variable will take up a
value lower than 111 with 50% probability or a value
higher than m with 5W/r probability again. In other
wonh. the median m splits up our prohahility
den~ity in exactly two equally probable parts. Since
the probability of occurrence of the random event
cquab the area under its demity rurve (i.e. the
definite integral under its density curve) then the
median 111 (of the lognormal distribution. in this
case) is defined as the integral upper limit m:
lln(11f-1,l·
2c,·
2 (83)
In order to find m, we may 1101 differentiate (83) with
respect to Ill , since the ··precise'" factor './i on the
right would then di~appear into a zero. On the
rontrary, we may try to perform the obvious
suh~titution
(in(,,)-μ)'
2 ()2
(84)
into the integral (83) to reduce it to the fulluwing
integral defining the error function crf(;:)
Random variable
Probability distribution
Probability density function
Mean value
Variance
Standard deviation
42
(85)
Then, after a few rcduetiom that we skip for the sake
of brevity, the full cquatiun (83) is turm;d intu
that i~
,,rf[ln(m)-;,)~o
✓2a
(86)
(87)
Since from the defimtion (85) one obviously has
erf(0J=0. (87) becomes
whence finally
ln(m)-p=O
✓2a
Irredian = 111 = e.u I.
(88)
(89)
This is the median of the log11ormaf distribution of
N. In other words, this is the 1111mber of
ExtraTerrestrial civiliza1io11s i11 1he Galaxy such
that, with 50% probability the acmal value of N will
be lower tha11 this media 11, a11d with 50% probabili1y
it will be higher.
In conclusion, we feel useful to summarize all the
equations that we derived about the random variable
N in the following Table 2.
N = number of communicating ET civilizations in Galaxy
Lognormal
/Jn/111--.uJ'-
fN(n)=I.. I lu' (II?'. 0)---,
n 5u
"-
(N)~,,"c'
,
a,\, 'p a'(a' i)= e- e e -1
a'
a,.v =cl-'e 2 Jerr' -1
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