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

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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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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 the correlation p€ (−1, +1). ρε (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. 1 (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]] 2 On the x² distribution.
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). ρε (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. 1 (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]] 2 On the x² distribution.
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