I Am Definitely Not Going to Get an Answer to This Question

Suppose I have a hat with X slips of paper in it. All the slips have a different number printed on them. I have no idea how many slips are in the hat. It could be one or it could be a trillion. I draw a slip out of the hat, and read the number, then put it back. Then I repeat Y times, until I draw a slip I have drawn previously. What is the expected value of X, given Y?

I tried to solve this with the probability equations I know, but I got nowhere. On one hand, it's probably unsolveable (If X may be any whole number, that gives us a big spoonful of infinity.) On the other hand though, similar problems *do* look solveable. For instance, consider the problem where I have the same hat, and just keep drawing numbers. Eventually I have drawn A times and have drawn A - B different numbers. Then (surely) we can say something about the expected value of X given A and B. (e.g. if B is zero and A is large then X is most likely A, right?). Or maybe not and I am tricking myself.

Would it help, in the original problem, if we said X was definitely less than a trillion, and equally likely anywhere within that range? Probably, but for real world applications, that's unrealistic, and we certainly don't always know what the distribution of X is. So my gripe is: Drawing the first repeat number after Y tries certainly seems to tell us *something*. So how can we walk away with no more of an idea about X than we started with?