Prove that y is also gaussian for any a ne 0 - what values

Question

Suppose X is a Gaussian random variable with mean Suppose we form a new random variable according to Y= aX +b  for constants a and b .

(a) Prove that Y is also Gaussian for any a ≠ 0 .

(b) What values for the constants a and b will lead to the new random variable Y having zero mean and unit variance?

 (c) What values for the constants a and b will lead to the new random variable Y having a mean of