Consider a random variable $X$ with continuous probability density $f(x)$ and compact support, say $[a,b]$ with $a<b$. Moreover, let $f(x)$ vanish at the boundary, i.e. $f(a) = f(b) = 0$.
Question:
What is the maximum variance which such a random variable can take?
EDIT:
It should have been mentioned here, that this problem is already solved in literature for probability densities without any further restrictions and the solution is given by a disrete (non-continuous) probability with weight 1/2 at the boundary. See comments and the "duplicate-link". However, I'm asking for a continuous probability density instead, which is zero at the boundary.
With these two additionnal conditions, the question is completely different. Maybe, a solution does not exist. Then, maybe there is a sequence of continuous probability densities $f_n(x)$, $n=1,2,3...$, whose limes has maximum variance. But I don't know how to construct such a sequence which gets peaked at the boundary.
If the distribution have a mean value $\mu$ and variance $S^2$, given the compact support $r=[a,b]$, the Popoviciu inequality states: $$S^2\le\dfrac{r^2}{4}$$ while the Bathia - Davies bound is: $$S^2\le (b-\mu)(\mu-a)$$