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]]
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]]
Transcribed Image Text: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]]
On the x² distribution.
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![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]]
On the x² distribution.](https://content.bartleby.com/qna-images/question/b41de797-8c36-43f3-a49e-0d77bbbd163e/051d8c8d-7aff-4f01-9b78-5e9ba0f7ef1f/5wqkej_processed.jpeg)
