4. For each of the following pairs of random variables X and Y, the conditional distribution of Y given {X = :r} is a named distribution, with parameters that may depend on :t. For each pair: i. \"trite down the conditional distribution of Y given {X 2 at}, including parameter(s). {2 marks each) ii. Find *1(Y|X). {I mart: each) iii. Find {[Y). {I mart: each) iv. Find Cov(X. Y). (2 marks each) You do not need to nd the marginai distributions of X and Y, which might not he named distrihatians. (a) Choose a number between 1 and 10, uniforrnly at random. and call that number X (i.e.. 1 g X 5 10). Take a standard deck of 5'2 cards and lay them out in a row, face up. Using a pen, put a checkmark on the rst X cards. Then turn the cards face down. shufe thoroughly, and draw 5 cards. Let Y be the number of marked cards you draw. [6- marl-cs] (b) Let N m DU[80), and let X have the value 5= 103 203 or 40 with probability 1/4 each. independent of N. If N > X . repeatedly subtract X from N until the result is X or smaller. Let Y be the number left over after this repeated subtraction. The number Y is aimast the same as the remainder teft over after dividing N into X equal parts. except that Y mitt eq-aat X, not 0, if N is evenly divisible by X. For erampte. consider the case X = 20. RN 2 77 then Y = 17'; if instead N = 60 then Y = 20; and if N219 then Y = 19. [6- marl-cs]