Uniform Distribution
Among probability distributions
which are nonzero over a
finite range of values
, the maximum-entropy
distribution is the uniform distribution. To show this, we
must maximize the entropy,
![]() |
(D.33) |
with respect to


Using the method of Lagrange multipliers for optimization in the presence of constraints [86], we may form the objective function
![]() |
(D.34) |
and differentiate with respect to


![]() |
(D.35) |
Setting this to zero and solving for

![]() |
(D.36) |
(Setting the partial derivative with respect to

Choosing
to satisfy the constraint gives
, yielding
![]() |
(D.37) |
That this solution is a maximum rather than a minimum or inflection point can be verified by ensuring the sign of the second partial derivative is negative for all

![]() |
(D.38) |
Since the solution spontaneously satisfied

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Exponential Distribution
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