Inner product of random Gaussian vector

[architect]

Hi,

I would like to ask a question please.

Assume we have a random vector X that is distributed under the Gaussian model and take the inner product of this vector and another constant vector d. Will the source distribution (Gaussian) remain the same? My intuition (although I might be wrong) says yes since the inner product may be regarded as a scaling operation. I know that this is true when we multiply a random variable with a constant, I just wonder if it applies to the inner product as well.

BR,

Alex.

 

[SW VandeCarr]

Hi,

I would like to ask a question please.

Assume we have a random vector X that is distributed under the Gaussian model and take the inner product of this vector and another constant vector d. Will the source distribution (Gaussian) remain the same? My intuition (although I might be wrong) says yes since the inner product may be regarded as a scaling operation. I know that this is true when we multiply a random variable with a constant, I just wonder if it applies to the inner product as well.

BR,

Alex.

By definition if a random vector x=x_1+x_2+...+x_n has a multivariate normal distribution, then the inner product y=<a,x>, where a is a constant vector:

y=a_{1}x_{1}+a_{2}x_{2},+...+a_{n}x_{n} is normally distributed.

 

[architect]

Thanks for your help. Appreciated!

BR,

Alex