The distribution for one of the components of a minimized sum? 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!
 A: The distribution of $X_1$ conditional on the event $[X_1+Y_1\lt X_2+Y_2]$ has density $g$ defined by
$$
g(x)=\frac1cf_X(x)u(x),
$$
with
$$
u(x)=\mathrm P(x+Y_1\lt X_2+Y_2),
\qquad
c=\int_{-\infty}^{+\infty} f_X(z)u(z)\mathrm dz.
$$
Since $u$ is nonincreasing, $g$ puts more weight than $f_X$ on the small values.
Note that the distribution of $\min\{X_1,X_2\}$ has density $m$, where
$$
m(x)=2f_X(x)v(x),\quad v(x)=\mathrm P(x\lt X_2)=1-F_X(x).
$$
A: I might write it this way.  Let $T = 1$ when $X_1 + Y_1 < X_2 + Y_2$, $2$ when $X_1 + Y_1 \ge X_2 + Y_2$.  You want the distribution of $X_T$.
Now $P(X_T < x|Y_1, Y_2) = P(X_1 < x, X_1 +Y_1 < X_2 + Y_2 )) + P(X_2 < x, X_1 + Y_1 \le X_2 + Y_2)$.  Suppose $X_1, X_2, Y_1, Y_2$ are all independent with continuous distributions (so I don't have to distinguish between $<$ and $\le$), and $|Y_2 - Y_1| = V $.
There are several different cases, depending on the ordering of $x$, $V$ and $1-V$.
For example, if $x < \min(V, 1-V)$, $P(X_T < x | Y_1, Y_2)$ is obtained by integrating $f_X(x_1) f_X(x_2)$ over a region that looks like this:

Then you'll have to integrate the result times $f_Y(y_1) f_Y(y_2)$ over the unit square to get unconditional probability.
