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Given the messy mail man problem of n pieces of mail, and X RV which value is "how many pieces of mail arrived to the right mail box," how do I compute E(X)?

I saw a solution using indicators, but I didnt really get it, so the more explanation or links to good sources in that subject would be appreciated.


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If the messy mailman distributes the mail independently at random, then this is the same problem as the expected number of derangements in a random permutation. – Austin Mohr Mar 5 '12 at 19:14
If the letters go independently, you don't have a permutation. And you mean the expected number of fixed points. – Robert Israel Mar 6 '12 at 0:24
up vote 4 down vote accepted

I suspect you're talking about a random permutation of $n$ letters. Let $X_i$ be the random variable that is $1$ if letter number $i$ is placed in the correct box, $0$ if not. Then $X$, the number of letters placed in the correct box, is $\sum_{i=1}^n X_i$. Since expected value is linear, $E[X] = \sum_{i=1}^n E[X_i]$. Now what is the probability that letter $i$ is placed in the correct box?

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