Question

Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2.

(a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.)

(b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y and the other is using the sum of two independent random variables.

(c) Calculate E(U) and V(U).

(d) Find the state space of U and calculate the cdf FU of U.

(e) Calculate Cov(U,X2) and ρ(U,X2).

Answer 1

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