# How can I check whether two signals are jointly normally distributed?

As explained on this Wikipedia page, if two random variables X and Y are uncorrelated and jointly normally distributed, then they are statistically independent.

I know how to check whether X and Y are correlated, but have no idea how to check whether they are jointly normally distributed. I hardly know any statistics (I learnt what a normal distribution is a couple of weeks ago), so some explanatory answers (and possibly some links to tutorials) would really help.

So my question is this: Having two signals sampled a finite number of N times, how can I check whether the two signal samples are jointly normally distributed?

For example: the images below show the estimated joint distribution of two signals, s1 and s2, where:

x=0.2:0.2:34;
s1 = x*sawtooth(x); %Sawtooth
s2 = randn(size(x,2)); %Gaussian


The joint pdf was estimated using this 2D Kernel Density Estimator.

From the images, it is easy to see that the joint pdf has a hill-like shape centred approximately at the origin. I believe that this is indicative that they are in fact jointly normally distributed. However, I would like a way to check mathematically. Is there some kind of formula that can be used?

Thank you.

• This is a simulation where you begin with signals that are not jointly normal by construction, and your statistical procedure seems to be showing that one can be reasonably confident that the signals are in fact jointly normal. So, should you be checking whether (a) the statistical method was applicable, or correctly applied, or correctly interpreted, or (b) your signal generation method is leading to signals that are in fact jointly normal even though a prima facie case cannot be made for joint normality (as would be the case if s1 = randn(size(x,2));; s2 = randn(size(x,2));?? – Dilip Sarwate Mar 12 '12 at 13:00
• @DilipSarwate That would be (b). I want a way to check whether the joint distribution is in fact normal. – Rachel Mar 12 '12 at 13:08