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Let $X_1,...,X_4$ be independent $N_p(μ,Σ)$ random vectors. Let $V_1,V_2$ be such that $$V_1=(1/4)X_1-(1/4)X_2+(1/4)X_3-(1/4)X_4 $$ $$V_2=(1/4)X_1+(1/4)X_2-(1/4)X_3-(1/4)X_4 $$

I need to find the marginal distributions of $V_1$ and $V_2$ and the joint density. Since they are linear combinations of random vectors I do not know the theory behind it to solve this. Any answers will be much appreciated. Thanks

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  • $\begingroup$ This looks like routine book-work - please read the link and modify your question and tags as needed. You should at least know the basic results about expectations, variances and covariances, and I presume you know that linear combinations of multivariate normals are multivariate normal. That should actually be sufficient here (though not automatically the simplest way to approach it). $\endgroup$
    – Glen_b
    Commented May 13, 2014 at 0:11
  • $\begingroup$ @Glen_b I should have mention that I only require how to compute the mean and co-variance I already knew it should be normal. $\endgroup$
    – Heisenberg
    Commented May 14, 2014 at 6:27
  • $\begingroup$ en.wikipedia.org/wiki/Expected_value#Linearity and en.wikipedia.org/wiki/Variance#Basic_properties $\endgroup$
    – Glen_b
    Commented May 14, 2014 at 6:39
  • $\begingroup$ @Glen_b thanks. I added a part from a text book in the answer below which also addresses the issue $\endgroup$
    – Heisenberg
    Commented May 14, 2014 at 6:42

2 Answers 2

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Use the following notation: Let $X$ be the 4 by $p$ matrix of your $X$ variables. Let $M$ be a 2 by 4 matrix of coefficients. Then $V=MX$ is your new variable. The distribution of $V$ is multivariate normal, with variance $M \Sigma M'$. To get the mean, apply the $M$ transformation to the means of the $X$ variable. Looks as if it should be 0.

Mardia's book on multivariate analysis is a good reference for this sort of problem.

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  • $\begingroup$ So what you are suggesting is to solve this the same way as say X is $N_p(μ,Σ)$ any linear combination of its elements are also normally distributed? Since this is a set of X's with $N_p(μ,Σ)$ what is the theory behind your answer? If it was the case for a linear combination of elements of one X then using the properties of multivariate normal distribution it can be solved. You have done the same if I am not wrong but how is that possible for random vectors? $\endgroup$
    – Heisenberg
    Commented May 12, 2014 at 14:06
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    $\begingroup$ The theory is that linear combinations of normals are normal. That's basically what you are doing here. $\endgroup$
    – Placidia
    Commented May 12, 2014 at 14:17
  • $\begingroup$ I have posted below a result that I found. Do you think this is the same as your answer? $\endgroup$
    – Heisenberg
    Commented May 13, 2014 at 14:58
  • $\begingroup$ is this the same? $\endgroup$
    – Heisenberg
    Commented May 14, 2014 at 6:26
  • $\begingroup$ Yes, that answers the question. Nice. $\endgroup$
    – Placidia
    Commented May 14, 2014 at 13:51
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enter image description here

I found this result from Johnson And Wichern's book. Adding this to the answers hoping that it would benefit someone

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