Gaussian Distribution

The Gaussian distribution has maximum entropy relative to all probability distributions covering the entire real line $x\in(-\infty,\infty)$ but having a finite mean $\mu$ and finite variance $\sigma^2$ .

Proceeding as before, we obtain the objective function

$$\begin{eqnarray*} J(p) &\isdef & -\int_{-\infty}^\infty p(x) \, \ln p(x)\,dx + \lambda_0\left(\int_{-\infty}^\infty p(x)\,dx - 1\right)\\ &+& \lambda_1\left(\int_{-\infty}^\infty x\,p(x)\,dx - \mu\right) + \lambda_2\left(\int_{-\infty}^\infty x^2\,p(x)\,dx - \sigma^2\right) \end{eqnarray*}$$

and partial derivatives

$$\begin{eqnarray*} \frac{\partial}{\partial p(x)\,dx} J(p) &=& - \ln p(x) - 1 + \lambda_0 + \lambda_1 x\\ \frac{\partial^2}{\partial p(x)^2 dx} J(p) &=& - \frac{1}{p(x)} \end{eqnarray*}$$

leading to

$$p(x) = e^{(\lambda_0-1)+\lambda_1 x + \lambda_2 x^2} = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2}{2\sigma^2}}.$$ (D.41)

For more on entropy and maximum-entropy distributions, see [48].