A teacher leaves out a box of N stickers for children to take home as treats. Children form a queue and look at the box one by one. When a child finds $k \geqslant 1 $ stickers in the box, he or she takes a random number of stickers that is uniformly distributed on $\{1,2,\dots,k\}$.
1- What is the expectation of the number of stickers taken by the second child, as a function of the initial number of stickers $N$?
2- If $E_N$ denotes the expected number of children who take at least one sticker from the box given that it initially contained N stickers. How can I compute a formula to represent $E_{N+1}$ in terms of $E_1 +\dots + E_N$. Also, how $E_N$ can be expressed in terms of $k$?
This looks like a variation of coupon collectors problem..http://en.wikipedia.org/wiki/Coupon_collector%27s_problem