$x_1,x_2,\dots,x_n$ are i.i.d RVs uniformly distributed on $\{1,2,\dots,N\}$. [duplicate]

Let $x_1,x_2,\dots,x_n$ be independent identically distributed random variables uniform on $\{1,2,\dots,N\}$, and let: $Y_n:=\text{the number of different elements in } \{x_1,x_2,\dots,x_n\}$.

Let $T:=\inf\{n:Y_n=N\}$.

What is $E\left[T\right]$?

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• What do you mean by $\inf\{n:Y_n=N\}$? – enthdegree Feb 10 '14 at 3:42
• The question is asking for the expected value of the random minimum sample size $T$ needed to be observed from a discrete uniform variable on $[1, N]$ such that every value in the support is observed. Clearly, ${\rm E}[T] \ge N$. – heropup Feb 10 '14 at 3:48
• @enthdegree the smallest n s.t. Yn=N.i.e. the first time we have N different elements in {x1,x2,...,xn}. – user98619 Feb 10 '14 at 3:50
• Get it now, thanks! – enthdegree Feb 10 '14 at 3:50
• @heropup You are right!Thank you for the explanation – user98619 Feb 10 '14 at 3:51

This question is commonly known as the coupon collector's problem. See here: http://en.wikipedia.org/wiki/Coupon_collector%27s_problem

• It is a good answer! – user98619 Feb 10 '14 at 4:18