Let $g \in \mathbb{R}^K$ be a vector of constants and let $x \in \mathbb{R}^K$ be a vector of Normal random variables with some means and variances. What is $E[\text{log}(g^Tx)]$? $$ E[\text{log}(g^Tx)]=\int\text{log}(g^Tx)f(g^Tx)dg^Tx $$
To proceed I need to know the distribution function of $g^Tx$. Wikipedia states that $g^Tx$ should be a univariate normal distribution with zero variance and a point mass on the mean, but I don't understand why that is.
Any insights would be appreciated! Or if there's a more straightforward way than the definition of the expectation.
Edit: as Estacionario points out, this is not well defined when x is negative. I have the additional constraint that $x$ is actually a truncated normal distribution with support only over $[0,1]$. For example, see https://en.wikipedia.org/wiki/Truncated_normal_distribution and set a=0 and b=1.
I don't think the expectation is well defined here. The linear combination of normal random variables $g^T x$ is itself normally distributed and hence can take negative values, of which you can't then take the logarithm.