Question

(4) (30 POINTS TOTAL) X and Y are discrete random variables; X has sample space 1,2} and Y has sample space 10, 1). Table 1 shows the joint distribution of (X, Y) TAbLe 1. Joint p.m.f. lC 4 (abcdef) Q搜索 (a) (5 points) Compute the marginal distribution of x and y, i.e. complete the following table 1 2 p(y) 1.3.4 p(x) (b) (5 points) Calculate the expectation of y, E[y] (c) (5 points) Calculate the conditional distribution of y, i.e. calculate p(ylr) for each value of x. Hint: you are looking for Pr(y-11x = 1), Pr(y-01x 11x 2), and Pr(y-01x = 2) 1), Pr(y (d) (5 points) Calculate the conditional expectation EIy.c] at each value of%, i.e. E[J1X 1 and E2] (e) (5 points) Calculate the expectation of the conditional expectation function over the distribution of : where i indexes the elements of the sample space of x. (f) (5 points) Compare your answers for parts (b) and (e). What is the theorem that characterizes this relationship?

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Answer 1

oluton o 02 nous tat thdishiou'on of ziz, p(1,1) 도 0.1 +0.3 : O'Y utMahot,te onginal iakibuion of 13 pts)