Let $f(x)$ be a positive continuous function on the real line that is the density of a random variable $X$. The differential entropy of $X$ is defined to be $-\int_{-\infty}^{\infty} f(x) \ln f(x) d x$. In which case does $X$ have the least differential entropy? You may use these facts: The differential entropy for a Gaussian with standard deviation $\sigma$ is $\ln (\sigma \sqrt{2 \pi e})$. The differential entropy of an exponential with mean $\lambda^{-1}$ is $1+\ln \left(\lambda^{-1}\right)$.
- $f(x):=(1 / 2) e^{-|x|}$.
- $f(x):=(\sqrt{100 \pi})^{-1} \exp \left(-|x|^{2} / 100\right)$.
- $f(x):=(\sqrt{20 \pi})^{-1} \exp \left(-|x|^{2} / 20\right)$.
- $f(x):=(1 / 4) e^{-|x| / 2}$.
- $f(x):=e^{-2|x|}$.