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If I am given two events $A$ and $B$ and the probabilities $P(A), P(A|B), P(B), P(B|A)$, and am told that the random variables $X$ and $Y$ are defined as

$$X(a) = 1\ \text{if}\ a \in A\text{, else}\ 0$$ $$Y(b) = 1\ \text{if}\ b \in B\text{, else}\ 0$$

how can I then go about to calculate the expected values of these variables? What probability density functions should I use for $p(a \in A)$ and $p(b \in B)$, given that none are specified explicitly?

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For any measurable set (event) $A$ the following holds: $E[1_A]=P(A)$. – Stefan Hansen Sep 1 '12 at 14:53
up vote 2 down vote accepted

Observe that $E[X]=\sum_{a\in A} X(a)p(a)=\sum_{a \in A} 1 \cdot p(a)= P(A)$.

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Similarly E[Y}=P(B). To answer your other question p(a∈A)=P(A) and p(b∈B)=P(B).

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