Among probability distributions
which are nonzero over a
*semi-infinite* range of values
and having a finite
mean
, the *exponential* distribution has maximum entropy.

To the previous case, we add the new constraint

(D.39) |

resulting in the objective function

Now the partials with respect to are

and is of the form . The unit-area and finite-mean constraints result in and , yielding

(D.40) |

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