Joint PMF of two order statistics with discrete parent distributions Let $X_1, X_2$ be i.i.d from a discrete distribution with finite support with cumulative distribution $F(x)$ and probability mass function $f(x)$. Let $X_{1:2}$ and $X_{2:2}$ represent the order statistics. How can I derive the joint probability function for $x<y$,
$$P[X_{1:2}=x,X_{2:2}=y]$$
of this two order statistics?
Note: I think it has to be the same as:
$$P[\textrm{min}[X_1,X_2]=x,\textrm{max}[X_1,X_2]=y]$$.

In the  $x < y$ case you get $P[X_{1:2}=x,X_{2:2}=y]$ $= P[X_{1}=x,X_{2}=y]+P[X_{1}=y,X_{2}=x] $ $= 2 P[X_{i}=x]P[X_{j}=y]$
while in the $x = y$ case you get $P[X_{1:2}=x,X_{2:2}=x] $ $=P[X_{1}=x,X_{2}=x] $ $=  P[X_{i}=x]^2$
and in the $x > y$ case you get $P[X_{1:2}=x,X_{2:2}=x] =0$
