Editing Chi-squared distribution (section)

=== Entropy ===
The [[differential entropy]] is given by
: <math>
    h = \int_{0}^\infty f(x;\,k)\ln f(x;\,k) \, dx
      = \frac k 2 + \ln \left[2\,\Gamma \left(\frac k 2 \right)\right] + \left(1-\frac k 2 \right)\, \psi\!\left(\frac k 2 \right),
  </math>
where <math>\psi(x)</math> is the [[Digamma function]].

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