3. Consider the univariate random variables X and Y with a joint density fx,y(x, y) = { ky(xy)(1x), 0 y x, 0 x 1, 0, otherwise, where k 0 is an unknown constant. (a) Evaluate k. (b) Write a simple R function for computing fx,y(x, y). (c) Write down fy|x(y|x). Hence find E(Y|X) and evaluate Var(E(Y|X)). (d) Find the covariance between X and Y. (e) Prove the law of total variance, i.e. Var(Y) E(Var(Y | X)) + Var(E(Y | X)). = [4 marks for each part; 20 marks in total]