# Distribution of the dot product of a multivariate gaussian random variable and a fixed vector

If $$a$$ is a multivariate normal random variable, and $$x$$ is a plain old vector (of the same shape as $$a$$), then the inner product $$x \cdot a$$ is a random variable. This post on math exchange suggests that the product will have a univariate normal distribution, but I haven't been able to frame the problem in a way that leads me to calculating $$\mu$$ and $$\sigma$$.

Edit: For posterity, the answer to this question was made much clearer to me by understanding a proof of the affine property.

The dot product is equivalent to linear combination in your situation. $$x\cdot a = x_1a_1 + x_2a_2+...+x_ka_k$$
Suppose $$a \sim N(\mu, \Sigma)$$, then scale $$x\cdot a$$ follows univariate normal distribution: $$x\cdot a\sim N(x\cdot \mu, x\Sigma x')$$
• Perhaps it is worth emphasizing that $x\cdot \mu$ is a scalar too, as is $x\Sigma x^\prime$ and thus $x\cdot a \sim N(\ldots)$ is referring to a univariate normal distribution – Dilip Sarwate Nov 2 '18 at 22:42
• Let $Y$ be random vector with mean $\mu$ and variance matrix $\Sigma$, $AY$ has the mean vector $A\mu$ and variance matrix $A\Sigma A'$ is true always. But the distribution of $AY$ depends on the distribution of $Y$. – user158565 Nov 4 '18 at 0:49