Question

Problem 1 Let Z be a random vector with expected value vector and covariance matrix E(Z) = [5; 4; 6] V(Z) = [3 2 1; 2 2 1; 1 1 1]. Define the random vector Y by Y1 = Z1 + 2Z3, Y2 = Z1 + Z2 - Z3, Y3 = 2Z1 + Z2 + Z3 - 5. 1. Find the expected value vector and covariance matrix of Y. 2. Find the expected value vector and covariance matrix of X = [Y1 + Y2 + 1; Y1 - Y2]. 3. Find the expected value and variance of W = 1/3(Y1 + Y2 + Y3).

          Problem 1
Let Z be a random vector with expected value vector and covariance matrix
E(Z) = [5; 4; 6] V(Z) = [3 2 1; 2 2 1; 1 1 1].
Define the random vector Y by Y1 = Z1 + 2Z3, Y2 = Z1 + Z2 - Z3, Y3 = 2Z1 + Z2 + Z3 - 5.
1. Find the expected value vector and covariance matrix of Y.
2. Find the expected value vector and covariance matrix of X = [Y1 + Y2 + 1; Y1 - Y2].
3. Find the expected value and variance of W = 1/3(Y1 + Y2 + Y3).

Problem 1
Let Z be a random vector with expected value vector and covariance matrix
E(Z) = [5; 4; 6] V(Z) = [3 2 1; 2 2 1; 1 1 1].
Define the random vector Y by Y1 = Z1 + 2Z3, Y2 = Z1 + Z2 - Z3, Y3 = 2Z1 + Z2 + Z3 - 5.
1. Find the expected value vector and covariance matrix of Y.
2. Find the expected value vector and covariance matrix of X = [Y1 + Y2 + 1; Y1 - Y2].
3. Find the expected value and variance of W = 1/3(Y1 + Y2 + Y3).

Added by Montserrat D.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

03/07/2023

Step 1

We have: E(Y) = [E(Y1), E(Y2), E(Y3)] = [E(Z1 + 2Z3), E(Z1 + Z2 - Z3), E(2Z1 + Z2 + Z3 - 5)] Using the linearity of expectation, we get: E(Y) = [E(Z1) + 2E(Z3), E(Z1) + E(Z2) - E(Z3), 2E(Z1) + E(Z2) + E(Z3) - 5] Now, let's find the covariance matrix of Y. We Show more…

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Problem 1: Let Z be a random vector with an expected value vector and covariance matrix E(Z) and V(Z), respectively. Define the random vector Y by Y1 = Z1 + 2Z3, Y2 = Z1 + Z2 - Z3, Y3 = 2Z1 + Z2 + Z3 - 5. 1. Find the expected value vector and covariance matrix of Y. 2. Find the expected value vector and covariance matrix of X = [Y1 + Y2 + 1; Y1 - Y2]. 3. Find the expected value and variance of W = 1/3(Y1 + Y2 + Y3).