Let $\mu$ be the mean and $\sigma$ the standard deviation of a probability distribution defined on the bounded interval $[a,b]$ (that is, the probability that the random variable lies outside $[a,b]$ is zero).
Does the following inequality hold generally for any such probability distribution?
$$\sigma^2 \le (\mu-a)(b-\mu)$$
Motivation: This inequality holds for the Beta distribution (as is obvious from the formulae here).
