- #1

- 5

- 0

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

multivariate normal distribution. If I compute the linear combination of X

and Y,

Z = aX+bY

I can compute the components of random vector Z.

Z = [aX_1+bY_1; aX_2+bY_2; ..... ;aX_n+bY_n]

In order to be a multivariate normal distribution, each component in random

vector should be a univariate normal distribution. In this point of view,

each component in random vector Z is a linear combination of univariate

normal distribution.(Because X and Y are multivariate normal distributions.)

Since the linear combination of univariate normal distribution is normal,

each component in random vector Z is also normal. Therefore Z, a linear

combination of multivariate normal distribution, is also normal.

----

My proof is right? If not, please tell me the wrong points. If right, then

how can I calculate the mean and covariance of the linear combination of

multivariate normal distribution.

Your comments are very helpful for me.

Thank you very much.