Inner product of random Gaussian vector

Main Question or Discussion Point

Hi,

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.

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Hi,

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.

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