linguistic:paraphrasing, structural:scholar_edition
Entropy

The differential entropy is given by

where ψ (x) {\displaystyle \psi (x)} is the Digamma function.

The chi-squared distribution is the maximum entropy probability distribution for a random variate X {\displaystyle X} for which E (X) = k {\displaystyle \operatorname {E} (X)=k} and E (ln (X)) = ψ (k / 2) + ln (2) {\displaystyle \operatorname {E} (\ln(X))=\psi (k/2)+\ln(2)} are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the log moment of gamma. For derivation from more basic principles, see the derivation in moment-generating function of the sufficient statistic.