Suppose X has a normal distribution with mean 0 and variance s^2. Let Y be an independent random variable taking values 1 and -1 with equal probability. Define Z= XY + X/Y. Which of the following is true?
(a) Var(Z) > s^2
(b) Var(Z) < s^2
(c) Var(Z) = s^2
(d) Var(Z) can be greater than or smaller than s^2
Variance of Y would be 1 and Var(Z) = Var(XY) + Var(X/Y) + 2 covariance(XY,X/Y). Can anyone suggest how to proceed further. Should I consider X and Y independent and then calculate?
Any help would be appreciated. Thanks.