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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
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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**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.
Transcribed Image Text:**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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