Paper 2, Section II, F

Let $X$ be a geometric random variable with $\mathbb{P}(X=1)=p$. Derive formulae for $\mathbb{E}(X)$ and $\operatorname{Var}(X)$ in terms of $p .$

A jar contains $n$ balls. Initially, all of the balls are red. Every minute, a ball is drawn at random from the jar, and then replaced with a green ball. Let $T$ be the number of minutes until the jar contains only green balls. Show that the expected value of $T$ is $n \sum_{i=1}^{n} 1 / i$. What is the variance of $T ?$

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