Let X be a random variable that takes on values between 0 and c. That is, P{0 ≤ X ≤ c} = 1.
Show that
Var(X) ≤ c2/4
One approach is to first argue that
E[X2] ≤ cE[X]
and then use this inequality to show that
Var(X) ≤ c2[α(1 − α)] where α =E[X]/c