For each one of the following statements, indicate whether it is true or false.
(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]
(a) False. If X = Y, it does not necessarily mean that the variance of X/Y is equal to zero. It depends on the distribution of X and Y.
(b) True. If X = Y, then the variance of X/Y is equal to the variance of X. This is because when X = Y, dividing X by Y will result in values of 1, which does not change the variability.
(c) True. When Y takes on the value y, the random variable Var(X/Y) takes the value E[(X-E[X/Y=y])^2/Y=y]. This is the expected value of the squared difference between the observed X and the expected value of X given Y=y.
(d) False. The correct expression for the random variable Var(X/Y) when Y takes on the value y is E[(X-E[X/Y=y])^2/Y=y]. The numerator should be subtracted by the expected value of X given Y=y, not the overall expected value E[X/Y].
(e) False. The correct expression for the random variable Var(X/Y) when Y takes on the value y is E[(X-E[X/Y=y])^2/Y=y]. The numerator should be subtracted by the expected value of X given Y=y, not the overall expected value E[X].
(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.