Paper 2, Section II, F

Probability | Part IA, 2009

I throw two dice and record the scores S1S_{1} and S2S_{2}. Let XX be the sum S1+S2S_{1}+S_{2} and YY the difference S1S2S_{1}-S_{2}.

(a) Suppose that the dice are fair, so the values 1,,61, \ldots, 6 are equally likely. Calculate the mean and variance of both XX and YY. Find all the values of xx and yy at which the probabilities P(X=x),P(Y=y)\mathbb{P}(X=x), \mathbb{P}(Y=y) are each either greatest or least. Determine whether the random variables XX and YY are independent.

(b) Now suppose that the dice are unfair, and that they give the values 1,,61, \ldots, 6 with probabilities p1,,p6p_{1}, \ldots, p_{6} and q1,,q6q_{1}, \ldots, q_{6}, respectively. Write down the values of P(X=\mathbb{P}(X= 2), P(X=7)\mathbb{P}(X=7) and P(X=12)\mathbb{P}(X=12). By comparing P(X=7)\mathbb{P}(X=7) with P(X=2)P(X=12)\sqrt{\mathbb{P}(X=2) \mathbb{P}(X=12)} and applying the arithmetic-mean-geometric-mean inequality, or otherwise, show that the probabilities P(X=2),P(X=3),,P(X=12)\mathbb{P}(X=2), \mathbb{P}(X=3), \ldots, \mathbb{P}(X=12) cannot all be equal.

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