8、2 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 thecorrelation p € (−1, +1).(a) Construct from X₁ and X2, a pair of random variables Z₁ and Z₂ whose joint distributionis the standard Gaussian distribution on R2, and such that X₁ = Z₁ and X₂ = aZ₁ +bZ2for constants a and b. Justify carefully that the standard Gaussian distribution on R² isindeed 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 thisrandom variable does not have a x² distribution. You may use the fact that E[Z₁] = 3.[Hint: first calculate E[X²X²]]On the x² distribution.

Given the two random variables and , whose joint distribution is Gaussian., where the correlation .

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] = 3. [Hint: first calculate E[X²X²]]