Suppose that X1 and X2 are two random variables whose joint distribution is Gaussian. Suppose that E[X1] = E[X2] = 0, that E[X1^2 ] = E[X2^2 ] = 1 and that E[X1X2] = where the correlation (1, +1).

(a) Construct from X1 and X2, a pair of random variables Z1 and Z2, whose joint distribution is the standard Gaussian distribution on R^2 , and such that X1 = Z1 and X2 = aZ1 + bZ2 for constants a and b. Justify carefully that the standard Gaussian distribution on R^2 is indeed the joint distribution of your choice of Z1 and Z2.

(b) Compute the variance of the random variable (X1^2 + X2^2) and deduce that if is not equal to 0 then this random variable does not have a ^2 distribution. You may use the fact that E[Z1^4 ] = 3. [Hint: first calculate E[X1^2X2^2 ] ]