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Let x be real valued random variable taking values on $a_1,\ldots, a_n$. Let $\Pr(x=a_i)=p_i$. Let $f$ be real valued function defined on $a_1, \ldots, a_n$

It is known that $$ E(f(x))=\sum_{i=1}^nf(a_i)p_i. $$

Would be the same formula true for $E(|f(x)|)$, i. e. $$ E(|f(x)|)=\sum_{i=1}^n|f(a_i)|p_i? $$

Thank you.

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Yes. As $|f|$ is "a real valued function, defined on ..." .. just apply your formula to $|f|$. – martini Jul 10 '12 at 20:24
up vote 1 down vote accepted

Let $ g(x) := |f(x)| $ be defined on $ a_1, \dots, a_n $. The composition of two real valued functions is a real valued function.

Now computing $E(g(x))$...

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