Question : Suppose that n balls are tossed into b bins so that each ball is equally likely to fall into any of the bins and that the tosses are independent.
c) What is the expected number of balls tossed until a particular bin contains a ball?
d) What is the expected number of balls tossed until all bins contain a ball?
My work:
For part (c), I defined a new indicator r.v $X_i$ which takes the value $1$ if all the previous $i$ balls didn't go into the bin and $0$ otherwise. Then it follows that the number of balls tossed until a particular bin contains a ball is:
$X=X_1+X_2+...+X_n+1$. Therefore $E(X)=1+\sum_{i=1}^n{E(X_i)}$. We know that $E(X_i)$ is the expectation that all i balls don't go into that particular bin so we have: $E(X_i)=(\frac{b-1}{b})^i$. So $E(X)=1+\sum_{i=1}^n{E(X_i)}=1+\sum_{i=1}^n{(\frac{b-1}{b})^i}=\sum_{i=0}^n{(\frac{b-1}{b})^i}=b(1-(\frac{b-1}{b})^n)$. Any problems with the answer so far?
(d) I am lost with this one, any hints on a suitable r.v?