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 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.(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] ]On 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. 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.

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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