# $k$-th order statistics when the value of $j$-th one is known

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)}

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

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 $$X_{(k:n)}$$ $$X_{(j:n)}$$