Question

0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X and Y? Are X and Y independent? Justify your answer. (d) Find the value of the pdf of Hence, or otherwise, estimate P(0.8 〈 X + y 〈 0.801). = X+ Y evaluated at u = 0.8. (e)

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