I was working a problem recently and the solution invoked the following identity for discrete random variables X and Y:
$$E[X | Y = 5] = \sum_{x \in X} x \, P[X = x | Y = 5 ] = \sum_{x \in X} P[X \geq x | Y = 5].$$
Is this so? They seem to refer to it as the ‘survival formula’. I googled but didn’t find anything and a proof is not immediate to me.
If it matters, X and Y were geometric. X was number of rolls of a die to get a 6 and Y was number of rolls to get an even number.