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

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7. Ifx is limited to a half line (p(x) = 0 for x <0) and the fin.t moment ofx is ihted at a:
a = f p(x dx
then the maximwn en opy OCCW1! when
p(x) =,! - f.r/al
a
and 1s equal to logea.
Now, we wish to point out that there is a third possible case, other than the two given
by Shannon. This is the case when the probability density function p(x) is limited to a
FINITE INTERVAL a::; x::; b. This is obviously the case with any physical POSITIVE
random variable, such as a distance, or the number N of extraterrestrial communicating
civilizations in the,". And it is easy to prove that for any such finite random variable the
maximum entropy distribution is the UNIFORM distribution over a -5, x -5, b. Shannon did
not bother to prove this simple theorem in his 1948 papers since he probably regarded
it as too trivial. But we prefer to point out this theorem since, in the language of the
statistical Drake equation, it sounds like:
"Since we don't know what the probability distribution of any one of the Drake random
variables D; is, it is safer to assume that each of them has the maximum possible
entropy over a; -5,x-5, h;, i.e., that D; is UNIFORMLY distributed there.
The proof of this theorem is along the same lines as for the previous two cases
discussed by Shannon:
We start by assuming that a; -5, x -5, b; .
We then form the linear combination of the entropy integral plus the normalization
condition for D;
where i is a Lagrange multiplier.
Performing the variation, one finds
- Iogp(x)- 1+1 = 0 that is: p(x)= e ,1,- i .
Applying the normalization condition (constraint) to the last expression for p(x) yields
b,
( )
f.b;
,l,-1
2-1 f.b;
,1,-1 (
)
I = f. p x dx =
e
dx = e
dx = e
b; - a,.
a1
a1
a1
that yields
,l,-1
1
e
= -­
h; - a;
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