Gaussian Distribution
The Gaussian distribution has maximum entropy relative to all
probability distributions covering the entire real line
but having a finite mean
and finite
variance
.
Proceeding as before, we obtain the objective function
![\begin{eqnarray*}
J(p) &\isdef & -\int_{-\infty}^\infty p(x) \, \ln p(x)\,dx
+ \lambda_0\left(\int_{-\infty}^\infty p(x)\,dx - 1\right)\\
&+& \lambda_1\left(\int_{-\infty}^\infty x\,p(x)\,dx - \mu\right)
+ \lambda_2\left(\int_{-\infty}^\infty x^2\,p(x)\,dx - \sigma^2\right)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2824.png)
and partial derivatives
![\begin{eqnarray*}
\frac{\partial}{\partial p(x)\,dx} J(p) &=& - \ln p(x) - 1 + \lambda_0 + \lambda_1 x\\
\frac{\partial^2}{\partial p(x)^2 dx} J(p) &=& - \frac{1}{p(x)}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2818.png)
leading to
![]() |
(D.41) |
For more on entropy and maximum-entropy distributions, see [48].
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