Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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;-;-s C 350-10 9 p;\s := Ks 0~S C I 10'
50
Ip C pip C Ip
100
IO
Olp C
100
ne ,Llne C ne one C
/3
50
fl C ,u.fl := fl
100
IO
ofl C
100
20
fi C ,Llfi := ti
100
IO
ofi C
100
20
fc ,u.fc = fc
100
IO
ofc C
100
10000
IL C ,ttL ·= tl.
1010 otL 1000
C
1010
:,;- .= "'.\"s Ip ne t1 fi fr tl. -=--· = 3500
Table 1. Input values (i.e. mean values and ~tandan.l dcviati(lm) for the seven Drake uniform random variables IJ,.
The first column (Ill the left list~ the seven input sheer numbers that alsn hccomc the mean values (middle cnlumn).
finally the last C(llumn nn the right lists the seven input stam.lan.l dcviatinns. The bottom line is the classical Drake
cquatiun ( I).
3.2 STEP 6: COMPUTING THE LOGS
OF THE 7 UNIFORMY
DISTRIBUTED DRAKE RANDOM
VARIABLESD;
Intuitively speaking, the natural log of a
uniformly distributed random variable may not be
another uniformly distributed random variable! This
is obvious from the trivial diagram of y = ln(x)
~huwn below:
Natural logarithm of x
, 2 3 4 5
POSITIVE independent variable x
Figure], The simpk functiun _,, = ln(xJ.
35
Su, if we have a uniformly di~tributcd random
variable D, with lowcr limit a, and uppcr limit b,, the
randmn variable
must have it~ range limited in between thc luwcr limit
fn(a,) and thc upper limit fn(b,J. In othcr word~, this
arc thc luwcr and upper limit~ of the relevant
probability den~ity function fl'. (.v). But what i~ the
actual analytic cxprc~sion of ~uch a pdf?. To find it,
wc mu~t rcsnrt tn thc gcncral tran~formation law for
random variable~, dcfincd by cquation (9). Herc we
ohviomly have
(24)
That, upon inversion, yields the single mot
(25)
On the other hand. differentiating (24) one gets
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