# the joint distribution of dependent random variables

Let $X \sim N(\mu_1, \sigma_1)$, $Y \sim N(\mu_2, \sigma_2)$, $Z \sim N(\mu_3, \sigma_3)$. I want to derive a joint distribution for $X/(X+Y+Z)$ and $Y/(X+Y+Z)$.

Since two random variables i.e. $X/(X+Y+Z)$ and $Y/(X+Y+Z)$ are dependent, I can not simply multiply them. So, how the pdf or cdf of their joint can be derived??

The joint CDF of $U = X/(X+Y+Z)$ and $V = Y/(X+Y+Z)$ is $F_{U,V}(u,v) = P(U \le u, V \le v)$ which can be obtained (in principle: I doubt that there's a closed form) by integrating the joint PDF of $X,Y,Z$ over the region $\{(x,y,z): x/(x+y+z) \le u, y/(x+y+z) \le v\}$. The joint pdf is then $f_{U,V}(u,v) = \dfrac{\partial^2}{\partial u \partial v} F_{U,V}(u,v)$.