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For this question I need to prove that $E(X^2)$ ≤ $E(X)$ given that X is a random continuous variable with probability distribution values between 0 and 1.
I also want to prove $Var(X)≤ 1/4$

So far I know that $Var(X)=E(X^2)-(E(X))^2$

Ben
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    The second part of your question ($Var(X)\leq 1/4$) is shown in http://stats.stackexchange.com/q/45588/40036. – josliber Nov 22 '16 at 21:59
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    For the first part, note that $X^2\leq X$ for your random variable, which pretty immediately gives $E(X^2)\leq E(X)$. – josliber Nov 22 '16 at 22:01

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