Suppose that X has a standard normal distribution. Let the conditional distribution of Y given X = x be normally distributed with mean E(Y | x) = 2x and variance V(Y | x) = 2x. Determine the following.

(a) Are X and Y independent? 

(b) P(Y < 3 | X = 3)

(c) E(Y | X = 3) 

(d) f_XY (x,y)

(e) Recognize the distribution f_XY (x, y) and identify the mean and variance of Y and the correlation between X and Y.