Given a r.v. $\tau$ and knowing that $ \Bbb{P}(\tau = \infty )> c > 0 $
1) I know that $\Bbb{E}(\tau) = \infty $ because $ \Bbb{E}(\tau) =\Bbb{E}(\tau ;\{\tau =+\infty \})+\Bbb{E}(\tau ;\{\tau < +\infty \} ) $
2) Now I would like to know if also $\Bbb{E}(\tau ;\{\tau < +\infty \} ) $ is infinite or not. So I thought about using the fórmula : $$ \Bbb{E}(\tau) = \int_0^{+\infty}{\Bbb{P}(\tau > t ) dt } $$
The question is: Does this fórmula compute $\Bbb{E}(\tau)$ or $\Bbb{E}(\tau;\{\tau < +\infty \} ) $ ?
Thanks a lot!
Max
The formula you have mentioned calculates $\mathbb{E}(\tau)$. For $\mathbb{E}(\tau;{\tau<\infty})$ you can do the following
$$ \mathbb{E}(\tau;{\tau<\infty})=\mathbb{E}(\tau\mathbb{1}(\tau<\infty))=\int_0^\infty\mathbb{P}(t<\tau<\infty)dt. $$