Let Z1 be N(m₁, σ²) and Z2 be N(m₂, σ²) Gaussian random variables (not
necessarily independent).
(a) Prove that X1 = Z1 + Z2, X2 = Z1 − Z2 are independent Gaussian
random variables.
I think you are supposed to assume that Z1 and Z2 have a joint bivariate normal distribution. Under that assumption, 2Z1-Z2 and Z1+Z2 have a joint bivariate normal distribution. I think you can compute the means, standard deviations, and correlation of 2Z1-Z2 and Z1+Z2 from the information given. Then you should be able to find the MGF from a known formula for the MGF of a bivariate normal.