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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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No:

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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    Oh no! I just realized that I misread. The book is http://www.proba.jussieu.fr/pageperso/amaury/index_fichiers/Guanajuato.pdf The page is 103. I thought that Fix would mean absorption (\tau<\infty) but it actually means absorption at 1.2012-04-24