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Give full justications for answers to Qlv] and to all parts of Q2 except Q2(viii). For Q1(i], Q1(ii], Ql(iii) and Q2(viii) only the answer is required. (1) Let X and Y be independent random variable; with distributions .1: 0 1 2 y 0 1 2 3 Pr(X=m) 41 % i Pr(Y=y) % g g 3 Let Z 2 XY. (i) Find E(Y). [Answer only required.] [1] (ii) Find Var(Y). [Answer only required.] [1] ) Write down a table giving the probability distribution for Z. [Answer only required.] [3] ) [iii (1v Are the random variables X and Z independent? [My justify your answer] [1] (2) Amy and Ben agree to play a \"best of 7\" backgammon match against each other. This involves playing games of backgammon until one player has won four games, at which point that player is declared the winner of the match and no more games are played. Suppose that each player has a 50% chance of winning each game, and that this is independent of whoever wins the other games. Let G be a random variable with a value equal to the number of games that are played in the match. Let X be the event that Amy wins the rst three games, let Y be the event that Amy wins the match and let Z be the event that G = 7. [N.B. The result of the match can be recorded as a string of A's and B's of length G which indicates who won each game. For example1 ABAAA would say that there were 5 games of which Ben only won the second one (so G = 5 and Amy won the match). Call this string the \"score\" for the match. You can write out all the possible scores, but there are a lot of them. It is better to classify and count scores without writing them all down. ] (i) Briey explain (without any calculation), why Pr(Y) = %. [1] (ii) Find Pr(Z). [2] (iii) Find Pr(Z | Y). [2] (iv) Find Pr(Y | X). [2] (v) Are X and Y independent events? [1] (vi) Are Y and Z independent events? [1] (vii) For g 6 {4,5,6, 7}, explain why Pr(G = g) is equal to [3] wer- (viii) Use the formula from part (vii) to nd E[G]. [Answer only required.] [2]