Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

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 $X,Y$.

share|improve this answer
    
I only know, that $X$ and $Y$ each are multivariat normal. I do not know the joint distribution. –  math12 May 27 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 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 at 19:39
    

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.