1) Suppose X ~ Bernoulli(p), but you don't know p (aside from the obvious that 0 <= p <= 1).

What is the largest possible value Var(X) can be?

2) Suppose that X and Y are i.i.d. Bernoulli(p) random variables (i.e., with the same value of p, 0 <= p <= 1). If Z = (X + Y)/2 What's the largest possible value Var(Z) can be?

3) Suppose X and Y are two random variables. You don't know their distribution, but someone tells you E[X] = E[Y]. Additionally, someone else (who are these people?) tells you that Var(X) < Var(Y). Which is smaller, E[X2] or E[Y2]?