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.
 A: Apart from graphical examination, you could use a test for normality. For bivariate data, Mardia's tests are a good choice. They quantify the shape of your distributions in two different ways. If the shape looks non-normal, the tests gives low p-values.
Matlab implementations can be found here.
A: This is more an extended comment than an effort to improve on the specific suggestion of @MånsT: Statistical test by and large are not tests for what distribution produced data but rather which ones did NOT. There are a few tests which are "tuned" to give answers to the normality question: Is this NOT from a Normal Distribution. The one sample Kolmogorov-Smirnov test is fairly widely known. The Anderson Darling test is perhaps more powerful in the one-D case. You should seriously ask yourself, WHY is the answer important? Often people ask the question for the wrong statistical purposes. Your example has demonstrated that your graphics-eyeball test has low power against an alternative composed of a sawtooth-Gaussian alternative, but you have not shown how that failure affects you underlying question.
