Timeline for Joint distribution of a random variable and the sample maximum

11 events

when toggle format	what		by	license	comment
Nov 27, 2023 at 14:56	history	edited	whuber♦		edited tags
Nov 27, 2023 at 14:55	answer	added	whuber♦		timeline score: 3
Nov 26, 2023 at 17:55	history	edited	kjetil b halvorsen♦	CC BY-SA 4.0	deleted 25 characters in body; edited tags
Mar 31, 2014 at 15:36	comment	added	Revin		Ah, so $$f_{X_1,X_{(n)}}(u,v)=\sum_i^n \frac{1}{n}f_{X_{(i)},X_{(n)}}(u,v)$$?
Mar 31, 2014 at 15:34	history	edited	Revin	CC BY-SA 3.0	added 16 characters in body
Mar 31, 2014 at 14:39	comment	added	whuber♦		Assuming the $X_i$ are exchangeable, for each $j$ there is a $1/n$ chance that $X_1$ is $X_{[j]}$. Thus the joint distribution of $(X_1,X_{[n]})$ is a mixture of the joint distributions of the $(X_{[j]}, X_{[n]})$. The rest is straightforward... .
Mar 31, 2014 at 4:32	comment	added	Revin		Would it be true that $f_{X_1,X(n)}=f_{X_1,X(n)\|X_1=X(n)}+f_{X_1,X(n)\|X_1\neq X(n)}$?
Mar 31, 2014 at 4:29	review	First posts
Mar 31, 2014 at 4:30
Mar 31, 2014 at 4:24	comment	added	Glen_b		One possibility is to consider two cases; first where $(n)=1$ (i.e. where $X_1$ is the biggest value) and then where $(n)\neq 1$.
Mar 31, 2014 at 4:23	comment	added	Dilip Sarwate		What if $X_1$ is the largest sample value?
Mar 31, 2014 at 4:12	history	asked	Revin	CC BY-SA 3.0