How do I compute the expectation of the following function with respect to a Multivariate Gaussian distribution?

Question

The function is as follows: $\boldsymbol{1}_{n}^{T} \exp\left(\boldsymbol{X\beta} \right).$ So the expectation that I want to compute is $\operatorname{E}_{\boldsymbol{\beta}} \left[\boldsymbol{1}_{n}^{T} \exp \left(\boldsymbol{X\beta} \right) \right]$ where $\boldsymbol{X}$ is a $n \times p$ matrix and $\boldsymbol{\beta}$ is a $p \times 1$ column vector and $\boldsymbol{\beta} \sim \mathcal{N}\left(\mu, \Sigma \right)$. ($\boldsymbol{1}_{n}^{T}$ is a $n \times 1$ column vector with $n$ ones and $\exp$ is an element-wise operator.)

Answer 1

$X \beta$ is also a normal distribution, with mean $X \mu$ and covariance $X\Sigma X^T$.

So, $[\exp(X \beta)]_i \sim LogNormal([X \mu]_i, [X \Sigma X^T]_{ii})$, and you can calculate out its expectation, call it $\xi_i$.

Then, $E[1^T \exp (X \beta) ] = \sum_i E \xi_i$ by linearity of expectation.