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I thought of this question and I somehow I can't figure out an answer

Let there be a box with $n$ balls (which have numbers from $1$ to $n$ to that you can distinguish them). When we randomly pick a ball we write down its number and put the ball back into the box. We repeat this untill we have picked out every ball at least once, let $A$ be the amount of picks. What is the expected value of $A$?

I have thought about certainly interesting question quite a while but I just can't manage to find an elegant solution, maybe someone has an idea?

Edit: This problem is equivalent to the Coupon Collector's Problem, which means that the correct solution is $n\sum_{k=1}^n\frac{1}{k}$

Svenson
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    This is a rewording of the Coupon Collector's Problem – lulu May 14 '16 at 16:57
  • To solve this problem beautifully you need to think it as a Markov chain, then you solve the expected value to change from state $n$ (after taking $n$ different numbers) to $n+1$ (i.e. the expected number of trials to take a new number when you already get $n$ numbers of $N$ different possible numbers). After you add all expectations (from $n=1$ to $n=N$) and violá! –  May 14 '16 at 17:02
  • This is a very similar problem (a bit more general) with the answer step by step. –  May 14 '16 at 17:09
  • Thanks for the phrase " Coupon Collector's Problem" that made it easier for me to find articles about this problem, I have successfull understood and solved it now :) – Svenson May 14 '16 at 17:43

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