I need help with this question. Thanks **Multivariate Gaussian Distributions****Given:**- \( X \) is multivariate Gaussian with: - Mean vector \(\mu_X = \begin{bmatrix} 6 & 0 & 8 \end{bmatrix} \) - Covariance matrix \(C_n = \begin{bmatrix} 1/2 & 1/4 & 1/3 \\ 1/4 & 2 & 2/3 \\ 1/3 & 2/3 & 1 \end{bmatrix} \)**Objective:**- Find the mean vector and covariance matrix of \( Y = \begin{bmatrix} y_1 & y_2 & y_3 \end{bmatrix} \), where: - \( y_1 = x_1 - x_2 \) - \( y_2 = x_1 + x_2 - 2x_3 \) - \( y_3 = x_1 + x_3 \)**Procedure:**1. **Mean Vector:** - Compute using the transformations defined for \( y_1, y_2, \) and \( y_3 \).2. **Covariance Matrix:** - Derive by applying the linear transformations to the covariance matrix \( C_n \).**Notes:**- The mean vector and covariance matrix help in understanding the distribution and interdependence of the variables in vector \( Y \).- These transformations are used frequently in statistical analyses and signal processing for converting distributions into desired forms.

Answered: Image /qna-images/answer/9fd34a6a-619b-4be3-b267-9f185eb7e187.jpg

X is multivariate Gaussian with μX = [6 0 8]and C₁ n = and covariance matrix of Y 1 = x1 - x2 Y2 = x1 + x₂ - 2x3 Y3 = x1 + x3 1/2 1/4 1/3 2/3 1/4 2 1/3 2/3 1 = [V₁ V₂ V3]- V2 32 Find the mean vector where

X is multivariate Gaussian with μX = [6 0 8]and C₁ n = and covariance matrix of Y 1 = x1 - x2 Y2 = x1 + x₂ - 2x3 Y3 = x1 + x3 1/2 1/4 1/3 2/3 1/4 2 1/3 2/3 1 = [V₁ V₂ V3]- V2 32 Find the mean vector where

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

I need help with this question. Thanks

**Multivariate Gaussian Distributions**

**Given:**
- \( X \) is multivariate Gaussian with:
- Mean vector \(\mu_X =
\begin{bmatrix}
6 & 0 & 8
\end{bmatrix}
\)
- Covariance matrix \(C_n =
\begin{bmatrix}
1/2 & 1/4 & 1/3 \\
1/4 & 2 & 2/3 \\
1/3 & 2/3 & 1
\end{bmatrix}
\)

**Objective:**
- Find the mean vector and covariance matrix of \( Y =
\begin{bmatrix}
y_1 & y_2 & y_3
\end{bmatrix}
\), where:
- \( y_1 = x_1 - x_2 \)
- \( y_2 = x_1 + x_2 - 2x_3 \)
- \( y_3 = x_1 + x_3 \)

**Procedure:**
1. **Mean Vector:**
- Compute using the transformations defined for \( y_1, y_2, \) and \( y_3 \).

2. **Covariance Matrix:**
- Derive by applying the linear transformations to the covariance matrix \( C_n \).

**Notes:**
- The mean vector and covariance matrix help in understanding the distribution and interdependence of the variables in vector \( Y \).
- These transformations are used frequently in statistical analyses and signal processing for converting distributions into desired forms.

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