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

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-e-£(,-)[E '(,}r+l]
(137)
.Ji;a .
,.2
Setting this derivative equal to zero means setting
E'(r)·r+l=O
(138)
That is, upon replacing (l 35) into (138), we get
Rearranging, this becomes
that is
(141)
whence
In[£] =!!_+ i!_
(142)
r
3
9
and finally
_!:!_
0'"2
r;mde"',.peak=Ce 3e 9
(143)
This is the most likely ET_Distance/rom Earth.
How likely?
To find the value of the probability density
function f ET_Disim,"' (r) corresponding to this value
of the mode, we must obviously replace () into ().
After a few rearrangements, which we skip for the
sake of brevity, one gets
Peak Value of fET_Distana,(r) = fET_Distana,(1rroctcl
... (144)
This is the peak height in the pdf f ET_Dist"""' (r) •
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:
J,"'
I
(145)
fET Distanu,(r)dr=­
0
-
2
Upon replacing ( 113), this becomes
(146)
2
In order to find m, we may not differentiate ( 146)
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
(147)
into the integral (146) to reduce it to the following
integral (85) defining the error function erf(z). Then,
after a few reductions that we leave to the reader as
an exercise, the full equation (145), defining the
median, is turned into the corresponding equation
involving the error function erj{x) as defined by (85):
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