Question

6. (15 points) Consider two independent Bernoulli r.v., U and V, both with probability of success 1/2. Let X = U+V and Y = |U - VI. (a) (5 points) Calculate the covariance of X and Y , oxy.

(b) (5 points) Are X and Y independent? Justify your answer.

(c) (5 points) Identify the random variable expressed as the conditional expectation of Y given X, i.e., E[Y |X].

Answer 1

a) Given that both U and V are independent here, therefore, the joint PDF for U, V here is obtained as:

P(U = 0, V = 0) = 0.5*0.5 = 0.25
P(U = 1, V = 1) = 0.5*0.5 = 0.25,
P(U = 1, V = 0) = 0.5*0.5 = 0.25,
P(U = 0, V = 1) = 0.5*0.5 = 0.25

From this, the joint PDF for X, Y here is obtained as:
P(X = 0, Y = 0) = P(U = 0, V = 0) = 0.25,
P(X = 2, Y = 0) = P(U = 1, V = 1) = 0.25,
P(X = 1, Y = 1) = P(U = 1, V = 0) + P(U = 0, V = 1) = 0.25 + 0.25 = 0.5

Therefore, the expected values here are computed as:
E(XY) = 1*P(X = 1, Y = 1) = 0.5,
E(X) = 2*0.25 + 1*0.5 = 1
E(Y) = 1*0.5 = 0.5

Therefore, Cov(X, Y) = E(XY) - E(X)E(Y) = 0.5 - 0.5 = 0

Therefore 0 is the required covariance here.

b) P(X = 0) = 0.25,
P(Y = 0) = 0.25 + 0.25 = 0.5

Therefore, P(X = 0, Y = 0) = 0.25 but this is not equal to P(X = 0)P(Y = 0)
Therefore X and Y are not independent variables here.

c) Given X, the conditional expected value of Y here is computed as:
E(Y | X = 0) = 0 because when X = 0, Y has to be 0.
E(Y | X = 1) = 1 because when X = 1, Y has to be 1
E(Y | X = 2) = 0 because when X = 2, Y has to be 0