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Suppose $X$ and $Y$ are normally distributed random variables.

I know that if $X$ and $Y$ are independent, then $(X,Y)$ is jointly normal.

But in general case, $X$ and $Y$ are not independent, $(X,Y)$ is not necessary jointly normal.

I wonder if there is a vector of affine-transformed random vectors, for example

$$Z_1 = a_1X+b_1Y$$

$$Z_2 = a_2X +b_2Y$$

$Z_1$ and $Z_2$ are surely normal, and also dependent.

Is $(Z_1, Z_2)$ jointly normal? If yes, how to prove it?

Zem
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    In the second half of your question (where you define $Z_1$ and $Z_2$), are $X$ and $Y$ assumed to be independent? – angryavian Jul 31 '21 at 20:23
  • Yes, exactly. $Z_1$ and $Z_2$ are linear combinations of independent normal random variables. – Zem Aug 01 '21 at 02:31

1 Answers1

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One of the equivalent definitions of "$(Z_1, Z_2)$ is jointly normal" is

$c_1 Z_1 + c_2 Z_2$ is normal for any choice of $c_1$ and $c_2$.

In your case, $c_1 Z_1 + c_2 Z_2$ can be rewritten in the form $c'_1 X + c'_2 Y$. Then using the fact that a linear combination of independent normal random variables is normal, you can conclude that $c_1 Z_1 + c_2 Z_2$ is normal.

angryavian
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