Paper 2, Section II, F

Probability | Part IA, 2013

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

A jar contains nn 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 TT be the number of minutes until the jar contains only green balls. Show that the expected value of TT is ni=1n1/in \sum_{i=1}^{n} 1 / i. What is the variance of T?T ?

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