Bound for the variance of a stochastic process

Question

Given a random variable $X$ and $N$ realizations of the stochastic process associated to $X$, a theorem gives a bound for the $\sigma^2[X]$: $$\sigma^2[X]\le\frac{1}{4}(A-a)$$ where $A$ and $a$ are given by: $$a\le X\le A$$ that means in $N$ realizations of the process, the random variable $X$ is bounded in that numeric interval. I tried to verify numerically this inequality, using normal and uniform distribution for $X$. The result is that it's verified but the difference between the real $\sigma^2$ and the $\frac{1}{4}(A-a)$ is not so little. My question is: is it possible to get a sharper bound for $\sigma^2[X]$ given $a$, $A$ and $X(k)$ for $1\le k\le N$? Thanks.

Answer 1

The bound is derived from the fact that you maximize the variance of a bounded random variable by assuming it only takes values of its bounds, and with equal probability. This is essentially a Bernoulli-type random variable, but with the lower and upper bounds replacing 0 and 1.

Since your question did not specify any particular structure for X, the answer is no. That bound cannot be made sharper because X could always be the "modified" Bernoulli process I just described, which would invalidate any sharper bound.