Question: i need help for this 4. Let (X, Y) be a pair of random variables having the joint density function f(x, y) = (Ta) '1{0

i need help for this

4. Let (X, Y) be a pair of random variables having the joint density function f(x, y) = (Ta) '1{0 0. (a) Let x be a fixed constant in (-1, 1). Show that given X = x, the conditional distribution of Y is uniform over some interval (-K, K). Specify K in terms of a and r. [4 marks] (b) Define a new pair of random variables (R, O) by X = Rcos(@) and Y = QRsin(0), with R E (0, 1) and O E [0, 27). It is known that the above transformation between (X, Y) and (R, O) is one-to-one. Find the joint density function of (R, O). [6 marks] (c) Deduce from (b) that Var ( VX2 + (Y/Q)? = 1/18. [5 marks]

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