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Let $X_1, X_2, X_3$, be jointly distributed according to a multivariate normal distribution.

$[X_1, X_2, X_3]^T\sim N(\mu=[0,0,0]^T , \Sigma = [[5,0,0],[0,2,1], 0,1,3]])$

$U = X_1 + 2X_2$ and $V = X_2 – X_3$

How to find the conditional distribution of $U$ given that $V = 1$.

gunes
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lsr729
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1 Answers1

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Both $U$ and $V$ are jointly normal RVs because this is a linear transformation over jointly normal RVs. So, by finding the mean, variance and covariance of these RVs you'll find the joint distribution of $U,V$.

Then, the conditional of one over another is also normal. You can find the related conditional using the joint distribution of $U,V$ as described here.

gunes
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  • How can I find the covariance between U and V? – lsr729 Mar 05 '20 at 07:05
  • just subtitute their definitions, you’ll have an expression in terms of X’s. Then, use distributive property of covariance and write it in terms of covariances between X’s. – gunes Mar 05 '20 at 07:59