I am having lots of trouble with a question that seems at firs quite elementary.
Let $X_1,X_2,\dots,X_n$ be independent and identically distributed random variables, $X_i:(\Omega,\mathcal{F},P')\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}),P_X)$ such that $P'$ determines a continuous cumulative distribution function $F_X(x)=P[X\leq x]=P'[\{\omega\in\Omega:X(\omega)\leq x\}]$ which is not necessarily absolute continuous wrt the Lebesgue measure. Then, given the order statistics $X_{i_1}<X_{i_2}<\dots<X_{i_n}$ the conditional distribution of $X_1,X_2,\dots,X_n$ is discrete, and assigns probability $\frac{1}{n!}$ to each "point" $X_{i_1},X_{i_2},\dots,X_{i_n}$ where $i_1.i_2,\dots,i_n$ is a permutation of $1,2,\dots,n$.
The questions does not seem to refer to the distribution of $X_1,X_2,\dots,X_n$ given a particular realization of the order statistics $X_{i_1}=x_{i_1},X_{i_2}=x_{i_2},\dots,X_{i_n}=x_{i_n}$, but the general conditioning.
I have found rather difficult to compute the conditional probabilities as they do not reduce to the simplest cases of conditional distributions.
Any insights or references?
Best Regards,
JM
This is an obvious statement (esp if you ignore the portion about absolutely continuous).
You should be able to show that $P(X_1 < X_2) = P(X_2 < X_1)$, which is the 2 variable case.
And similar, for any permutation $\sigma \in S_n$, we have
$$P(X_1 < X_2 < \cdots <X_n) = P(X_{\sigma(1)} < X_{\sigma(2)} < \cdots < X_{\sigma(n)})$$
Hence, each of these $n!$ events are equally likely to occur, hence occur with probability $\frac{1}{n!}$ (with the disclaimer that they sum to 1).
Note that you are restricting your cases to when all the order statistics are distinct, which means that all the original RV's have distinct values. This might restrict your probability space, if it is not absolutely continuous.
The exact distribution of $X_i$ does not matter. All that matters is that they are IID, and also that it is not the constant RV, in which case, $P(X_1< X_2) = 0$.