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The question goes like this: Let $A_1,A_2 , A_3 $ be events with probabilities $\frac15$, $\frac14, and \frac13$ respectively. Let N be the number of these events that occur. Write down a formula for N in terms of indicators.

I am confused on 2 parts. Firstly, I'm not sure what this problem is saying about the 3 events. Does it sound like it's saying the probabilities of $A_1,A_2 , A_3 $ happening all at the same time is the listed probabilities? And then secondly how would we come up with a formula for the number of times these events occur. I would say $N={1,2,3,...,N}$ but that cant be right.

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    $\begingroup$ The function $1_{A_i}$ is $1$ iff $A_i$ occurs. Hence $N = \sum_{k=1}^3 1_{A_k}$. To compute the expected value of $N$, we have $E N = E (\sum_{k=1}^3 1_{A_k}) = \sum_{k=1}^3 E 1_{A_k} = \sum_{k=1}^3 P(A_k)$. $\endgroup$ – copper.hat Oct 23 '12 at 4:34
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$$N=\mathbf 1_{A_1}+\mathbf 1_{A_2}+\mathbf 1_{A_3}$$

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  • $\begingroup$ As @copper.hat explained. $\endgroup$ – Did Oct 23 '12 at 5:00

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