Suppose I have a bag of `N` balls, each labeled with a distinct integer. The experiment starts when I draw a ball from the bag, record its label, and then return the ball back to the bag. I repeat this step until I draw a ball that I have already drawn before. Let `X` be the number of balls I drew in the experiment. Let `P` be the probability distribution of `X`.


Is there a name to this probability function `P`?


Is there a closed form for `P`?


Actually, I would be satisfied if there is a closed form for `E[X]` and `Var[X]`.


Please help.



Probability of first collision with replacement
$P(X)=(X-1)N^{1-X}\frac{(N-1)!}{(N+1-X)!}$, for $X=2,3,\ldots N+1$.


