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]]
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
Related questions
Question
8、2
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 44 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning