P.m.f., expected value, and variance of number of rolls of a fair die needed for every face to appear at least once.

Question

I think I have found that $\operatorname E(X) = 14.7.$

However, I'm not sure about the pmf or $\operatorname{Var}(X).$

I have the hint that $X = 1 + X_1 +X_2+\cdots+X_5$ and that for some reason $P(X_i = x) = (i/6)^{x-1}(1-(i/6))$

Can anyone explain why that would be the case & also how to find the variance.

Answer 1

This is the well known coupon collector problem. In particular let $X$ be the number of rolls needed for every face to appear once. Then $X=X_{1}+\dotsb++X_{6}$ where $X_1=1$ and $X_{i}$ is the waiting time needed to get a face distinct from the $i-1$ faces seen before for $i\geq 2$. In particular $X_{i}$ independent and geometrically distributed with probability of success $p_{i}=\frac{6-i+1}{6}$. It follows that $$ EX=\sum_{i=1}^6\frac{6}{6-i+1}=6\left(1+\frac{1}{2}+\dotsb+\frac{1}{6}\right) $$ while $$ \text{Var}(X)=\sum_{i=2}^6\frac{1-p_{i}}{p_{i}^2} $$ by independence.