Maximum Entropy Property of the Normal Distribution

The differential entropy of a continuous random variable $X$ with PDF $f$ is $h(X) = -\int_{-\infty}^{\infty} f(x) \ln f(x)\, dx$.

(a) Among all continuous distributions on $\mathbb{R}$ with mean $\mu$ and variance $\sigma^2$, use the method of Lagrange multipliers to show that the PDF maximizing $h(X)$ satisfies $\ln f(x) = -1 + \lambda_0 + \lambda_1 x + \lambda_2 x^2$ for some constants $\lambda_0, \lambda_1, \lambda_2$.

(b) Determine $\lambda_0, \lambda_1, \lambda_2$ by enforcing the three constraints ($\int f = 1$, $\int xf = \mu$, $\int x^2 f = \mu^2 + \sigma^2$) and show that $f$ is the $N(\mu, \sigma^2)$ PDF.

(c) Compute $h(X)$ for $X \sim N(\mu, \sigma^2)$.