Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$

Question

Let T be a non-negative integer-valued random variable with $\mathbb{E}(T) < \infty $. Prove that $\mathbb{E}(T) = \sum^\infty_{k=1}\mathbb{P}(T \geq k)$.

Had a few attempts, haven't really got anywhere. I'm wondering as I'm typing this if proof by induction is a good way to go.

Edit: One major thing I forgot to add, am I correct in thinking that also, $\mathbb{E}(T) = \sum^\infty_{k=1}k\mathbb{P}(T = k)$?

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score 7 · Accepted Answer · answered Feb 02 '14 at 01:48

7

$$T=\sum_{k=1}^\infty\mathbf 1_{T\geqslant k}=\sum_{k=1}^\infty k\cdot \mathbf 1_{T=k}$$

answered Feb 02 '14 at 01:48

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Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$

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