Thread: Probability question normal random variables

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.

Hint: See if you can compute the covariance of X1 and X2.

I thought that independant variables have zero covariance but zero covariance does not imply independance. In which case is it sufficient to show that cov(x1,X2)=0?

For normally distributed random variables, zero covariance implies independence.

ok thanks I didn't know that was the case.

I am not sure how to approach part b of this question.
b)if cov(Z₁,Z₂)=ρσ²
Find the M.G.F of (X₁,X₂)=(2Z₁-Z₂,Z₁+Z₂)

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.

Thanks i figured it out. It was basically what you stated.