Thread: Show that X+Y is normal

If X and Y have a bivariate normal distribution with correlation p, show that X +Y is normally distributed


This seems like a pretty standard proof but cant find it anywhere. Simply adding the marginal distributions, X+Y is what I tried but how is the correlation coefficient introduced?
I then looked at convolution but that didnt seem to go the right way either...

Originally Posted by Gekko

If X and Y have a bivariate normal distribution with correlation p, show that X +Y is normally distributed


This seems like a pretty standard proof but cant find it anywhere. Simply adding the marginal distributions, X+Y is what I tried but how is the correlation coefficient introduced?
I then looked at convolution but that didnt seem to go the right way either...

Use the change of variable theorem:

Make the transformation U = X + Y, V = X - Y. The inverse transformation is , . Then .

Get the joint pdf of U and V. Integrate wrt to v to get the marginal, which is the required pdf.

By definition of a multivariate normal distribution, if a random vector follows it, say (X,Y), any linear combination of its components follows a normal distribution, X+Y for example.

Thanks for your reply.

How do we handle the standard deviations of x and y when converting to U and V? Additionally, taking this approach means we start from the marginal distributions of X and Y. Isnt there a way to start from the joint bivariate distribution of X and Y since this includes the correlation term?

Unfortunately this approach doesnt seem to work. I end up with a very messy exponential which doesnt allow separation for the marginal calculation

Any thoughts? Is this not a standard proof?