Expectations of stopping times in general

Question

I have a very basic question: So for a stopping time $\tau$ with $E(\tau)<\infty$ we have $E(\tau)=\sum_{n=0}^\infty P(\tau>n)$, right? Why is that? Thanks!

Answer 1

This is an usual "trick" :

By definition,

$$E(\tau) = \sum_{n=0}^\infty n P( \tau = n)$$

But as

$$n = \sum_{i=0}^{n-1} 1 $$

you get

$$E(\tau) = \sum_{n=0}^\infty \left( \sum_{i=0}^{n-1} 1 \right) P( \tau = n) = \sum_{n=0}^\infty \left( \sum_{i=0}^{n-1} P( \tau = n) \right)$$

Now, you invert both sums

$$E(\tau) = \sum_{i=0}^\infty \sum_{n=i+1}^{\infty} P( \tau = n)$$

And as the events $\{\tau = n\}$ are disjoints, you have that

$$\sum_{n=i+1}^{\infty} P( \tau = n) = P( \tau > i )$$

Hence the result :

$$E(\tau) = \sum_{i=0}^\infty P( \tau > i )$$