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 thecorrelation p€ (−1, +1).ρε(a) Construct from X₁ and X2, a pair of random variables Z₁ and Z₂ whose joint distributionis the standard Gaussian distribution on R², and such that X₁ = Z₁ and X₂ = a Z₁ +bZ₂for constants a and b. Justify carefully that the standard Gaussian distribution on R2 isindeed 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 thisrandom variable does not have a x² distribution. You may use the fact that E[Z₁] = 3.[Hint: first calculate E[X²X₂]]2On the x² distribution.

Given the two random variables and , whose joint distribution is Gaussian., where the correlation .

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.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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