What are the distributions of these Gaussian variables?

Question

I'm just wondering if I have the correct answers to these questions.

Let $X$, $Y$, and $Z$ be multivariate Gaussian distributed with mean vector and covariance matrix: $$ \mu = (0,1,2)^T, \quad \Sigma = ((4,5,1),(5,9,-2),(1,-2,1)) $$ What is the distribution of $(Z,X)^T$?

I have $$ (Z,X)^T \sim ( (2,0)^T, ((1,1),(1,4)) ) $$ Calculate the distribution of $(Z,X)^T | Y = 4$.

I have $$ (Z,X)^T | Y = 4 \sim ( (11/3,-2/3)^T, ((-16/9,19/9),(19/9,32/9)) ) $$ What is the distribution of $2X - 3Y$? I have $$ 2X-3Y \sim (-3, -65) $$

Answer 1

• The first one is correct.

• The second one follow this link. But you can tell it is net correct as the diagonal elements of the covariance matrix can NOT be negative.

• The last one follows: $$ aX + bY \sim N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2 + 2ab\sigma_{X,Y}) $$ so, it is not correct. BTW, variance can NOT be negative.