1. Let Z = (Z1, Z2, Z3) be a vector with i.i.d. N(0, 1) components. Let...

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Question

1. Let Z = (Z1, Z2, Z3) be a vector with i.i.d. N(0, 1) components. Let...

1. Let Z = (Z1, Z2, Z3) be a vector with i.i.d. N(0, 1) components.

Let r be a constant with 0 < r < 1. Define X1 = √ rZ1 + √ 1 − rZ2 and X2 = √ rZ1 + √ 1 − rZ3.

(a) Give the distribution of X1 and the distribution of X2. Find Cov(X1, X2).

(b) Give the matrix A so that the vector X = (X1, X2) is a transform X = AZ. Give the distribution of X and its covariance matrix.

math Statistics-And-Probability

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Answer 1

Answer #1

1.

(a)

According to the question,

&

Then,

Therefore ,

&

Now,

Also,

Then,

So,

(b)

Define,

and

Then , A is 2 x 3 matrix.

i.e.

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Answer 2

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