The discrete random variable $Y$ has distribution given by

$$\mathbb{P}(Y=k)=(1-p)^{k-1} p, \quad k=1,2, \ldots$$

where $p \in(0,1)$. Determine the mean and variance of $Y$.

A fair die is rolled until all 6 scores have occurred. Find the mean and standard deviation of the number of rolls required.

[Hint: $\left.\sum_{i=1}^{6}\left(\frac{6}{i}\right)^{2}=53.7\right]$