1
$\begingroup$

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: $$P[\textrm{min}[X_1,X_2]=x,\textrm{max}[X_1,X_2]=y]$$.

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ Hint: for a given pair (x,y) either $(X_1,X_2)=(x,y)$ or $(X_1,X_2)=(y,x)$ $\endgroup$
    – Xi'an
    Commented Sep 8, 2021 at 10:41
  • $\begingroup$ Sorry I dont get it :/ $\endgroup$
    – oyy
    Commented Sep 8, 2021 at 10:52

1 Answer 1

Reset to default
2
$\begingroup$

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 $P[X_{1:2}=x,X_{2:2}=x] =0$

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.