Does anyone know what distribution the sum of two normal distributions will have if they are not independent, but have co-variance a?
If it is needed, they can have the same mean, but I would rather not have this condition, if possible.
I have been looking for this, but I can only find it for independent RVs.
The question is related to question 4c) of this:
http://www.maths.ox.ac.uk/filemanager/active?fid=4675
This is the solution sheet, but in his proof of 4c) he appears to have said that two non-independent normal distribtions sum to give a normal distribution with the variance shown (1). (If you are right about two correlated normal distributions, then this is fine).
But he has then said by (2) that cov(X,Y)=0 implies independence. I did not think this was true. Any idea why he would say such a thing?
Oli,
You have to be careful how you phrase your question.
If X and Y have a joint normal distribution then their sum has a normal distribution. However, if X and Y each have a normal distribution then it is not necessarily the case that they have a joint normal distribution, and then the result does not follow.
See the Wikipedia entry on the multivariate normal distribution,
Multivariate normal distribution - Wikipedia, the free encyclopedia
especially the section titled "A Counterexample".
The same web page explains why the sum is normally distributed if X and Y have a joint normal distribution; see the section titled "Affine Transformation".
It is also true that if X and Y have a joint normal distribution and are uncorrelated then they are independent. The Wikipedia entry says so but does not give a proof. However, a proof is fairly simple. If you look at the joint pdf when the correlation is zero, you will see that it factors into the product of a function of X times a function of Y, which shows that X and Y are independent.
[Edit]Be careful: If X and Y do not have a joint normal distribution, then being uncorrelated does not imply their independence.