Question

Say $\boldsymbol{\beta}$ is a random n-vector having the multivariate normal distribution with mean $\boldsymbol{b}$ and covariance matrix $\boldsymbol{S}$. And let $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$ be two row vectors with n elements each. Then we know the distribution of $\frac{exp(\boldsymbol{x_1\beta})}{1+exp(\boldsymbol{x_1\beta})}$ and $\frac{exp(\boldsymbol{x_2\beta})}{1+exp(\boldsymbol{x_2\beta})}$. Then can we determine the joint distribution of these two random variables?

Answer 1

If $A$ is the matrix with rows $x_1$ and $x_2$, then $A \beta$ has a bivariate normal distribution with mean $A b$ and covariance matrix $A S A^T$. From that you can get the joint distribution of your two random variables.