Find the joint distribution - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:08:44Z http://mathoverflow.net/feeds/question/84275 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84275/find-the-joint-distribution Find the joint distribution Chao 2011-12-25T17:51:06Z 2011-12-25T21:09:53Z <p>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?</p> http://mathoverflow.net/questions/84275/find-the-joint-distribution/84283#84283 Answer by Robert Israel for Find the joint distribution Robert Israel 2011-12-25T21:09:53Z 2011-12-25T21:09:53Z <p>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.</p>