Coupon collector waiting-time variance ($\mathrm{Var}(T)=n^2{\sum }_{r=1}^n{r}^{-2}-n{\sum }_{r=1}^n{r}^{-1}$)

For uniform independent draws of $n$ coupon types, the waiting times with $r$ unseen types remaining are independent positive-support geometric distributions of parameter $r/n$. Summing their means gives $n{\sum }_{r=1}^{n}1/r$, and summing their variances gives the displayed formula. Independence follows from fresh draws after each discovery and the fact that the next stage's conditional distribution depends only on the count of remaining types.