Does it matter here that random variables are jointly normally distributed?

Question

My lecture notes ask the following (true/false) question on understanding:

Jointly normally distributed random variables are independent iff they are uncorrelated.

I don't quite understand what role does the word jointly play here. As far as I know:

$$(X,Y) \text{ is normal} \iff \text{both } X \text{ and } Y \text{ are normal}.$$

So from what I can tell, the statement would still be true if we removed the word jointly. Is there any need to emphasis it here?

Answer 1

The brilliant example of normal but not jointly normal uncorrelaeted but dependent variables is here: http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent Briefly if $X$ is normal and $Y=WX$ where $W$ has binary $-1,1$ distribution then $X,Y$ both are normal uncorrelated but dependent. They are not jointly normal.

Answer 2

First of all, jointly distributed means that there exists a density function that depends on both x and y. Moreover, that will be a bivariate normal density function.

Now, for the normal distribution, the equivalence (iff) is indeed true but the proof is a little tricky. The question in your notes asks about the equivalence between a lack of correlation and independence. Note that in general, independence always implies uncorrelatedness. The converse is not always true.

So the forward direction is straightforward to see. Why should there be some correlation between two quantities if they are realized indepedently?

The reverse direction is slightly trickier. It has to do with the fact that the variance-covariance matrix for a bivariate, uncorrelated system can be written with zeroes as the off-diagonal elements, which makes the uncorrelated case indistinguishable from the independent case. Since the normal distribution only depends on the first and second moment (and the variance-covariance matrix captures the second moments which matter in your case), we can conclude that independence and uncorrelatedness are indeed equivalent for two jointly normal variables.

Lastly, a diagonal variance covariance matrix also means that the joint density of x and y can be written as the product of the densities, i.e. $f(x,y) = f(x)\cdot f(y)$.