Question: Consider a fair tetrahedral dice, whose faces are numbered by 1, 2, 3, 4, and which is thrown twice. Let X be the sum and

Consider a fair tetrahedral dice, whose faces are numbered by 1, 2, 3, 4, and which is thrown twice. Let X be the sum and Y the maximum of the two respective numbers on the faces falling downside.

(a) Find the joint distribution P ı .X; Y /1 of X and Y .

(b) Construct two random variables X0 and Y 0 on a suitable probability space . 0;F0; P0/, which have the same distributions as X and Y (i.e., P ıX1 D P0

ıX01 and P ıY 1 D P0

ı Y 01), but so that the distribution of X0 C Y 0 differs from that of X C Y .

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