How do i prove that E[R^2] is greater than or equal to (E[R])^2??

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- Jul 21st 2009, 10:18 AMPink LadyProving an equality
How do i prove that E[R^2] is greater than or equal to (E[R])^2??

- Jul 21st 2009, 01:50 PMLaurent
This is related to the variance: on one hand, we define $\displaystyle {\rm Var}(R)=E[(R-E[R])^2]$, which is (obviously?) nonnegative. On the other hand, you should expand the above expression in order to show that $\displaystyle {\rm Var}(R)=E[R^2]-E[R]^2$. Since this is nonnegative, you must have $\displaystyle E[R^2]-E[R]^2\geq 0$, and this is it. So you need to justify why the first expression of $\displaystyle {\rm Var}(R)$ is nonnegative, and why it equals the second one.

(There would be another more advanced/general answer involving convexity (and Jensen inequality), but you don't need that. )