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Part BCSIR NET June 2023

Multivariate normal distribution

Let $X = (X_{1}, X_{2})^{T}$ be bivariate normal with mean (0, 0)ᵀ and covariance $\sum = [[5, - 3], [- 3, 10]]$ . The mean vector and covariance matrix of $Y = (X_{1}, 5 - 2 X_{2})^{T}$ are

A.(0, 5)ᵀ, [[5, −3], [−3, 40]]
B.(0, 5)ᵀ, [[5, −6], [−6, 20]]
C.(0, 5)ᵀ, [[5, 3], [3, 20]]
D.(0, 5)ᵀ, [[5, 6], [6, 40]]✓

Solution

$E [5 - 2 X_{2}] = 5; V a r (5 - 2 X_{2}) = 4 \cdot 10 = 40; C o v (X_{1}, 5 - 2 X_{2}) = - 2 \cdot (- 3) = 6$ .

Related counterexample: If every marginal is normal then the vector is multivariate normal

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From Linear Models and Multivariate › Multivariate normal distribution

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