Maximizing Variance

X is an r.v. from the interval [0,1]. What's the maximum variance for X?


Solution:

Intuitively, you'll guess it's 1/4--just binomial from {0,1}. The proof:

Instead of looking at the interval [0,1], let's look at the inter [-1/2, 1/2]. This does not change the variance properties. We have Var[X] = E[X^2] - E[X]^2 <= E[X^2] <= 1/4. So the upper bound is 1/4. Can the upper bound be reached? Sure, take bionomial from {-1/2, 1/2}.