The way to represent the multivariate normal distribution like that comes from the book named 'Mathematical statistics'(Peter J.Bickel, Kjell A. Doksum). At the chapter B,(It may be the appendix.) they represent the bivariate normal distribution like this.
X = a_{11}u_{1} + a_{12}u_{2} +...
I'm sorry. The expression U_{1} and U_{2} is wrong. I want to represent the two dependent random variable X and Y using a standard normal distribution u_{1} and u_{2} .
I'm sorry again. Now, the X and Y are multivariate normal distributions which dimension is d. So, I have to...
Thank you very much, statdad and Stephen Tashi. :)
Okay. Then how about this idea?
First, in order to prove the linear combination of dependent univariate normal distribution, I write the two random variables in this way.
X = a_{11}u_{1} + a_{12}u_{2} + \mu_{x}
Y = a_{21}u_{1} +...
How can I prove that the any linear combination of multivariate normal
distribution is also normal?
I can prove it but I'm not sure that this is right or not. The point of my
proof is as follows.
---
The X and Y has the same dimensional random vector, and each random vector is...