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I mostly see stopping times used to stop a random process at time $\tau$. However, in some proofs, the following identity is used, which I cannot deduce myself:

Let $\tau$ be a stopping time. Why $$ E(\tau) = \sum _{i=1}^\infty P(\tau \geq i). $$ $\tau$ is a discrete stopping time, so why not $$ E(\tau) = \sum _{i=1}^\infty iP(\tau = i). $$

laurensvm
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    This is called the tail sum formula, and is true for ANY random variable $\tau$ taking non-negative integer values. Hint: $\tau=\sum_{k=1}^\infty\mathbf1(\tau\geq k)$ – jlammy Jan 22 '21 at 21:27
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    This identity is true for all (weakly) positive integer-valued random variables. See this question: https://math.stackexchange.com/questions/843845/find-the-mean-for-non-negative-integer-valued-random-variable – charlus Jan 22 '21 at 21:28

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