Let $X_1,…,X_n$ be i.i.d. random variables. Are these two equalities correct?
$P[X_{(2)}<x_2,…,X_{(n)}<x_n| X_{(1)}=x_1]=\\ =n!P[X_{2}<x_2,…,X_{n}<x_n| X_{1}=x_1]=\\ =n! P[X_{2}<x_2]…P[X_{n}<x_n]$
And if not, why? I am a bit confused as I don’t know if the permutations are also considering the fact that the minimum $X_{(1)}$ could be any of the $n$ variables.
Thank you.