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Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$.

Suppose also that we know one of the realization, say $X_{(j)}=x_{(j)}$, and we also know that it is the $j$-th lowest value among $\{x_1,\cdots,x_n\}$.

In this case, what would be the revised order statistics of the $n$ values?

Formally, what would be a closed form representation of

$$Prob[X_{(k:n)}<z|X_{(j:n)}=x_1],~k\neq j$$

where $X_{k:n}$ represents the $k$-th order statistics among $n$ samples.

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  1. Write the joint density of $(X_{(k:n)},X_{(j:n)})$ as detailed on the order statistics Wikipedia page.
  2. Write the marginal density of $X_{(j:n)}$ as detailed on the order statistics Wikipedia page.
  3. Apply Bayes' formula for the conditional density of $X_{(k:n)}$ $X_{(j:n)}$
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