Question

Let y be a 3-dimensional multivariate normal random vector with mean and variance ? = [3 0 -2]^T, V = [[2, 0, 0], [0, 2, 0], [0, 0, 1]]. Let A = 1/10 [[4, -2, 0], [-2, 1, 0], [0, 0, 10]]. (a) Describe the distribution of Ay. (b) Find E[y^T Ay]. (c) Describe the distribution of y^T Ay. (d) Find all linear combinations of y elements which are independent of y^T Ay.

          Let y be a 3-dimensional multivariate normal random vector with mean and variance
? = [3 0 -2]^T, V = [[2, 0, 0], [0, 2, 0], [0, 0, 1]].
Let
A = 1/10 [[4, -2, 0], [-2, 1, 0], [0, 0, 10]].
(a) Describe the distribution of Ay.
(b) Find E[y^T Ay].
(c) Describe the distribution of y^T Ay.
(d) Find all linear combinations of y elements which are independent of y^T Ay.

Let y be a 3-dimensional multivariate normal random vector with mean and variance
? = [3 0 -2]^T, V = [[2, 0, 0], [0, 2, 0], [0, 0, 1]].
Let
A = 1/10 [[4, -2, 0], [-2, 1, 0], [0, 0, 10]].
(a) Describe the distribution of Ay.
(b) Find E[y^T Ay].
(c) Describe the distribution of y^T Ay.
(d) Find all linear combinations of y elements which are independent of y^T Ay.

Added by -Ngeles L.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Supreeta N

08/13/2023

Step 1

Since $y$ is a multivariate normal random vector, the linear transformation of $y$ is also a multivariate normal random vector: $$ Ay \sim N(A\mu, A\Sigma A^T) $$ Show more…

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Let y be a 3-dimensional multivariate normal random vector with mean and variance. Let Describe the distribution of Ay. Find E[yT Ay]. Describe the distribution of yT Ay. Find all linear combinations of y elements which are independent of yT Ay.