start with the definition
E(N) = \sum_{k=1}^{\infty}kP(N=k)= \sum_{k=1}^{\infty}\sum_i^kP(N=k)
= \sum_{i=1}^{\infty}\sum_k^{\infty}P(N=k)
= \sum_{i=1}^{\infty}P(N\ge i)
tex isn't working for me lately, here
Hi all,
I had this problem as part of an example in class. I really didn't understand what was happening so I wondered if anybody could enlighten me as to what's going on here. Any help is greatly appreciated.
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same idea, but I tried to avoid a triple sum
\sum_{i=1}^{\infty} i P(N\ge i)
=\sum_{i=1}^{\infty} i \sum_{k=i}^{\infty}P(N=k)
=\sum_{k=1}^{\infty}\sum_{i=1}^k i P(N=k)
now use \sum_{i=1}^k i =k(k+1)/2
=\sum_{k=1}^{\infty}{k(k+1)\over 2} P(N=k)
=E(N(N+1))/2