# Sum of normal distributions

• Apr 10th 2008, 04:32 AM
Oli
Sum of normal distributions
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.
• Apr 10th 2008, 05:17 AM
CaptainBlack
Quote:

Originally Posted by Oli
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 sum will be normally distributed. The mean will be the sum of the means,
and to find the variance:

V=E([(x1+x2)-(mu1+mu2)]^2)=E( [(x1-mu1)+((x2-mu2)]^2)

Now expand this and simplify.

RonL
• Apr 10th 2008, 05:31 AM
Oli
Do you know why they will be normally distributed?

I can see that is how to find the mean and variance if they are.

Thankyou very much.
• Apr 10th 2008, 05:40 AM
Oli
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?
• Apr 10th 2008, 05:49 AM
mr fantastic
Quote:

Originally Posted by Oli
[snip]
But he has then said by (2) that cov(X,Y)=0 implies independence.[snip]

This statement is completely wrong. Independence => Cov = 0, but Cov = 0 does NOT => independence.
• Apr 10th 2008, 06:11 AM
awkward
Quote:

Originally Posted by Oli
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.

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.

Be careful: If X and Y do not have a joint normal distribution, then being uncorrelated does not imply their independence.
• Apr 10th 2008, 07:18 AM
Oli
Think that sounds exactly right, thanks awkward.
• Apr 10th 2008, 10:54 AM
CaptainBlack
Quote:

Originally Posted by mr fantastic
This statement is completely wrong. Independence => Cov = 0, but Cov = 0 does NOT => independence.

But it does for bivariate normals (because the covariance matrix is diagonal
and so joint distribution comes apart as the product of two normal densities
in the two variables)

RonL