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Does knowing that for a iid $X_j$'s, $EXj = 0$ and $EX_j^2 < \infty$ indicate $E|X_j|$ is finite too? How to show that.

Can we say: $EX_j^2 = E|X_j|^2 < \infty \rightarrow \text{then } E|X_j| < \infty$ I appreciate your help.

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    You can use Jensen's inequality for $|X|$, for example: $E|X|^2 - (E|X|)^2 \geq 0$ to create an upper bound for $E|X|$2012-11-29
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    Cool...thanks...2012-11-29
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    No need for jensen's inequality. Use $Var(|X|) \geq 0$. Its the same thing but you get it from here also.2012-11-29

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