# Finding conditional distribution when matching ordering

Suppose we draw two values $$x_1,x_2$$ according to a CDF $$F$$. Independently, we draw another two values $$y_1,y_2$$ according to another CDF $$G$$. Both $$F$$ and $$G$$ has support $$[0,1]$$.

Among those four values, I first observe $$x_1$$ only. And then, I get to observe one of $$y_1$$ and $$y_2$$, depending on whether $$x_1\leq x_2$$ or $$x_1>x_2$$ following the rule below:

--- If $$x_1\leq x_2$$, then I observe $$y_1$$ or $$y_2$$, whichever is smaller (or equal to) than the other.

--- If $$x_1>x_2$$, I observe $$y_1$$ or $$y_2$$, whichever is greater than the other.

So, if my $$x_1$$ is smaller than the other $$x$$, my observation of $$y$$ equals the smaller value of $$y_1$$ and $$y_2$$. If my $$x_1$$ is larger than the other $$x$$, the $$y$$ I observe is the larger one between $$y_1$$ and $$y_2$$.

In this case, if I observe $$x_1$$ and some value $$y$$, what is the conditional distribution of the other value of $$y$$? (so, if $$y=y_1$$, then what is the distribution of $$Y_2$$?)

We assume that $$F$$ is continuous, so that the probability of $$x_1=x_2$$ is 0.
Let $$y_u$$ be the unobserved value of $$y$$, with $$y_o$$ the observed value. Then the cdf for $$y_u$$ is $$\begin{cases} F(x_1)\dfrac{G(y_u)}{G(y_o)} &\text{ if }y_u \le y_o\\ \\ 1 - \big(1-F(x_1))\dfrac{1-G(y_u)}{1-G(y_o)} &\text{ if }y_u \ge y_o \end{cases}$$ Note that this properly gives:
• a cdf of $$0$$ when $$y_u=0$$
• a cdf of $$1$$ when $$y_u=1$$
• a cdf of $$F(x_1)$$ when $$y_u=y_o$$ (calculated from either side).
The last statement reflects that the probability of $$y_u is exactly the probability of $$x_2.
• Thank you for your comment @Matt F. When $y_0=1$, i.e., the upper case, the CDF is not well defined. For example, if $F$ is uniform and $x_1=1/2$, the CDF for $y_u$ becomes $\frac{y_u}{2}$.. Mar 3, 2021 at 8:22
• I agree this isn’t well-behaved for $y_o=1$; I think the reasonable area of concern is $0<y_o<1$. Mar 3, 2021 at 9:15