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I saw a book where they calculate $E(\tau 1_{\tau<\infty})$ and $E(\tau)$ for some random variable $\tau$ (actually a stopping time of a process). They obtain different results. The problem is that for this variable; $P(\tau<\infty)=1$. Thus in my mind I would expect $E(\tau 1_{\tau<\infty})=E(\tau)$.

Can you give an example of a random variable $\tau$ which satisfies both of the following properties?

1) $P(\tau<\infty)=1$.

2) $E(\tau 1_{\tau<\infty})\neq E(\tau)$

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up vote 2 down vote accepted


If $\mathrm P(\tau\lt\infty)=1$, then $\tau=\tau\mathbf 1_{\tau\lt\infty}$ almost surely. Since $X=Y$ almost surely implies that $\mathrm E(X)=\mathrm E(Y)$, this implies that $\mathrm E(\tau)=\mathrm E(\tau\mathbf 1_{\tau\lt\infty})$.

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Which book? $ $ – Did Apr 24 '12 at 16:14
Thanks! I will add the info about the book and quote it in the question. – wircho Apr 24 '12 at 16:16
Oh no! I just realized that I misread. The book is The page is 103. I thought that Fix would mean absorption ($\tau<\infty$) but it actually means absorption at 1. – wircho Apr 24 '12 at 16:27

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