Q 8.2. Suppose that X₁ and X2 are two random variables whose joint distribution is Gaussian. 1 and that E[X₁ X₂] = p where the Suppose that E[X₁] E[X₂] = correlation p € (-1, +1). - = = 0, that E[X2] = E[X²] = (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 X1 + 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.
Q 8.2. Suppose that X₁ and X2 are two random variables whose joint distribution is Gaussian. 1 and that E[X₁ X₂] = p where the Suppose that E[X₁] E[X₂] = correlation p € (-1, +1). - = = 0, that E[X2] = E[X²] = (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 X1 + 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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8.2
![Q 8.2. Suppose that X₁ and X₂ are two random variables whose joint distribution is Gaussian.
0, that E[X²] = E[X²] = 1 and that E[X₁X₂] = p where the
=
Suppose that E[X₁] = E[X₂]
correlation p E (−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 = a Z₁ +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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2F8f95db3f-1d97-46a7-abec-75a55f7cf71c%2Fmn1va3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q 8.2. Suppose that X₁ and X₂ are two random variables whose joint distribution is Gaussian.
0, that E[X²] = E[X²] = 1 and that E[X₁X₂] = p where the
=
Suppose that E[X₁] = E[X₂]
correlation p E (−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 = a Z₁ +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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