I have a complicated thing I would like to find the distribution for.
Let's say I have a random variable $X\sim F_X(x)$ supported over $(0,1)$. I have two independent draws from $F_X$, which are $x_1$ and $x_2$.
Similarly, I have a random variable $Y\sim F_Y(y)$ supported over $(0,1)$. I have two independent draws from $F_Y$, which are $y_1$ and $y_2$.
I am not even sure what words to use to express this properly, but I want to find the distribution for (or any way to express) the "$x$" part of the $Minumum[x_1 + y_1, x_2 +y_2]$. That is probably not clear enough so let me try to explain more:
If $x_1 + y_1 < x_2 +y_2$ I want some way of expressing a distribution for $x_1$ that is more specific than $F_X$ since now we have more information about it. (And if $x_1 + y_1 > x_2 +y_2$ the of course I would want the way to describe $x_2$).
Does this question make sense? And if so, can anyone help me, even if it's just to have better terminology for describing what I'm looking for?
Thanks so much!