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.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 4E
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