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Question

Assume that the bivariate Gaussian random vector X = [X₁, X₂]ᵀ (denotes transpose) has the following mean vector and covariance matrix: μ = [1, 2]ᵀ Σ = [2.5, 2; 2, 4.1] 1.1 (2 pts) Compute the correlation coefficient ρ (rho) between X₁ and X₂. 1.2.1-2 (4 pts) Determine the eigenvalue matrix Λ = diag(λ₁, λ₂) of Σ with the eigenvalues sorted from large to small. 1.3.1-4 (8 pts) Determine the corresponding eigenvector matrix U of Σ with U₁₁, U₁₂, U₂₁, U₂₂ where the first components of the two eigenvectors are chosen to be nonnegative. 1.4.1-2 (4 pts) Determine the mean vector for the probability density function of X, i.e., m = [m₁, m₂].

          Assume that the bivariate Gaussian random vector X = [X₁, X₂]ᵀ (denotes transpose) has the following mean vector and covariance matrix:
μ = [1, 2]ᵀ
Σ = [2.5, 2; 2, 4.1]

1.1 (2 pts) Compute the correlation coefficient ρ (rho) between X₁ and X₂. 
1.2.1-2 (4 pts) Determine the eigenvalue matrix Λ = diag(λ₁, λ₂) of Σ with the eigenvalues sorted from large to small. 
1.3.1-4 (8 pts) Determine the corresponding eigenvector matrix U of Σ with U₁₁, U₁₂, U₂₁, U₂₂ where the first components of the two eigenvectors are chosen to be nonnegative. 
1.4.1-2 (4 pts) Determine the mean vector for the probability density function of X, i.e., m = [m₁, m₂].

Added by Richard V.

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