Question: Consider two random variables X and Y with joint PMF given in Table 5.5 Define the random variable Z as Z = E[X|Y ]. a.

Consider two random variables X and Y with joint PMF given in Table 5.5 Table 5.5: Joint PMF of X and Y in Problem 13 X=0 X = 1 Y = 0 6 1 Y = 1 1 Y = 2

Define the random variable Z as Z = E[X|Y ].

a. Find the Marginal PMFs of X and Y .

b. Find the conditional PMF of X, given Y = 0 and Y = 1, i.e., find P_X|Y (x|0) and P_X|Y (x|1).

c. Find the PMF of Z.

d. Find EZ, and check that EZ = EX.

e. Find Var(Z).

Table 5.5: Joint PMF of X and Y in Problem 13 X=0 X = 1 Y = 0 6 1 Y = 1 1 Y = 2

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