Question

2. The random variables $X_1$, $X_2$ and $X_3$ are independent, with $X_1 sim N(0,1)$, $X_2 sim N(1,4)$ and $X_3 sim N(-1,2)$. Consider the random column vector $X = [X_1, X_2, X_3]^T$. (a) Write $X$ in the form $X = mu + BZ$, where $Z$ is a vector of iid standard normal random variables, $mu$ is a 3 imes 1 vector, and $B$ is a 3 imes 3 matrix. (b) What is the covariance matrix of $X$? (c) Determine the expectation of $Y_1 = X_1 + X_3$. (d) Determine the distribution of $Y_2 = X_1 + X_2 - 2X_3$. (e) Give the joint distribution of $Y_1$ and $Y_2$.

          2. The random variables $X_1$, $X_2$ and $X_3$ are independent, with $X_1 sim N(0,1)$, $X_2 sim N(1,4)$ and $X_3 sim N(-1,2)$. Consider the random column vector $X = [X_1, X_2, X_3]^T$.
(a) Write $X$ in the form
$X = mu + BZ$,
where $Z$ is a vector of iid standard normal random variables, $mu$ is a 3 	imes 1 vector,
and $B$ is a 3 	imes 3 matrix.
(b) What is the covariance matrix of $X$?
(c) Determine the expectation of $Y_1 = X_1 + X_3$.
(d) Determine the distribution of $Y_2 = X_1 + X_2 - 2X_3$.
(e) Give the joint distribution of $Y_1$ and $Y_2$.

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2. The random variables X1, X2 and X3 are independent, with X1 sim N(0,1), X2 sim N(1,4) and X3 sim N(-1,2). Consider the random column vector X = [X1, X2, X3]^T.
(a) Write X in the form
X = mu + BZ,
where Z is a vector of iid standard normal random variables, mu is a 3 imes 1 vector,
and B is a 3 imes 3 matrix.
(b) What is the covariance matrix of X?
(c) Determine the expectation of Y1 = X1 + X3.
(d) Determine the distribution of Y2 = X1 + X2 - 2X3.
(e) Give the joint distribution of Y1 and Y2.

Added by Stacey B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

09/17/2023

Step 1

Since X1, X2, and X3 are independent normal random variables, we can write Z as a vector of independent standard normal random variables, i.e., Z = [Z1; Z2; Z3] where Z1, Z2, and Z3 are iid N(0,1). Now, let's find p and B: p = [E(X1); E(X2); E(X3)] = [0; 1; Show more…

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The random variables X1, X2, and X3 are independent, with X1 ~ N(0,1), X2 ~ N(1,4), and X3 ~ N(-1,2). Consider the random column vector X = [X1, X2, X3]^T. (a) Write X in the form X = μ + BZ, where Z is a vector of iid standard normal random variables, μ is a 3 × 1 vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Y1 = X1 + X3. (d) Determine the distribution of Y2 = X1 + X2 - 2X3. (e) Give the joint distribution of Y1 and Y2.

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