# Can $E(\tau 1_{\tau<\infty})$ be different from $E(\tau)$ if $P(\tau<\infty)=1$?

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 $\mathrm P(\tau\lt\infty)=1$, $\tau=\tau\mathbf 1_{\tau\lt\infty}$ almost surely; since $X=Y$ almost surely implies that $\mathrm E(X)=\mathrm E(Y)$, this implies that $\mathrm E(\tau)=\mathrm E(\tau\mathbf 1_{\tau\lt\infty})$.
Which book?  –  Did Apr 24 '12 at 16:14
Oh no! I just realized that I misread. The book is 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. –  wircho Apr 24 '12 at 16:27