Q 8.2. Suppose that X₁ and X2 are two random variables whose joint distribution is Gaussian. Suppose that E[X₁] = E[X₂] = = 0, that E[X²] E[X₂] = 1 and that E[X₁X₂] = p where the correlation p€ (-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 R2, and such that X₁ = Z₁ and X₂ = aZ₁ +bZ2 for constants a and b. Justify carefully that the standard Gaussian distribution on R² is indeed the joint distribution of your choice of Z₁ and Z₂. (b) Compute the variance of the random variable X2 + X2 and deduce that if p = 0 then this random variable does not have a x² distribution. You may use the fact that E[Z₁] [Hint: first calculate E[X²X2]] = 3.