For each one of the following statements, indicate whether it is true or false.

Question

(a) If X = Y (i.e., the two random variables always take the same values), then var(X/Y) = 0

(b) If X= Y (the two random variables always take the same values), then var(X/Y) = var(X)

(c) If Y takes on the value y, then the random variable Var(X/Y) takes the value
E[(X-E[X/Y=y])^2/Y=y]

(d) If Y takes on the value y, then the random variable Var(X/Y) takes the value
E[(X-E[X/Y])^2/Y=y]

(e) If Y takes on the value y, then the random variable Var(X/Y) takes the value
E[(X-E[X])^2/Y=y]

Answer 1

(a) False. If X and Y always take the same values, then the variance of their ratio would be undefined since division by zero is not allowed.

(b) True. If X and Y always take the same values, then the variance of their ratio would be the same as the variance of X (since the ratio would always be 1).

(c) True. The expression represents the conditional expectation of the squared difference between X and the conditional expectation of X given Y=y, where the expectation is taken over all possible values of X given Y=y.

(d) False. The expression represents the conditional expectation of the squared difference between X and the overall expectation of X given Y (not specific to the value y), where the expectation is taken over all possible values of X given Y=y.

(e) False. The expression represents the conditional expectation of the squared difference between X and the overall expectation of X (not specific to the value y), where the expectation is taken over all possible values of X given Y=y.