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Let $X\sim N(\mu_1,V_1),~~Y\sim N(\mu_2,V_2)$. How can I show that $X$ and $Y$ are independent?

I am wondering how I can show this.

I only know the following case: $Z=(Z_1,\ldots,Z_n)\sim N(\mu_3,V_3)$: Then $Z_i$ are independent if $\text{cov}(Z_i,Z_j)$ for all $i\neq j$.

But here the situation is different, because $X$ and $Y$ are both multivariate normal distributed. Indeed I do not know how to show the independence in this case. Can you help me?

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up vote 2 down vote accepted

If you know that $(X,Y)$ are jointly normal, i.e. the full vector $[X',Y']'$ has a normal distribution then all you need to do is check the covariance of $X,Y$ i.e. check whether

$$E(XY') - E(X) E(Y')=0.$$

If you only know that $X$ and $Y$ are separately multivariate normal then you need to check whether the joint distribution function equals the product of the marginals, but you didn't give us the joint distribution function of $X,Y$.

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I only know, that $X$ and $Y$ each are multivariat normal. I do not know the joint distribution. – math12 May 27 '14 at 19:35
Then you're up the proverbial creek without a paddle. Please note that that would be equally true if $X,Y$ were scalar-valued. – JPi May 27 '14 at 19:37
I guess it is my mikstake... and that there actually is a joint distribution but I do not see. In my task, $X$ is the estimator for residuum, and Y is the least square estimator. Maybe there is a connection I do not see. – math12 May 27 '14 at 19:39

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