Expected value of characteristic function

Question

I was wondering about the expected value of the characteristic function $\chi_A$ for some set $A \in \mathcal{A}$ in a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. It should be $\mathbb{E}(\chi_A) = \mathbb{P}(A)$, but I can't figure out why that is the case.

Thanks for any help!

Do you know how to compute the expectation for "simple functions"? — Calculon, Apr 28 '15 at 11:55

score 1 · Answer 1 · answered Apr 28 '15 at 12:01

1

Hint:

If $X$ is a random variable taking values in a countable set $S$ then: $$\mathbb EX=\sum_{s\in S}s.\mathbb P(X=s)$$

$\chi_A$ is actually a random variable taking values in $\{0,1\}$.

answered Apr 28 '15 at 12:01

drhab

151,093

Oh thanks! Got it now – Steven Apr 28 '15 at 12:09

Expected value of characteristic function

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