Given a set of n iid random variables $\{X_i\}_{i=1}^n$, it is not hard to find the probability function (via the CDF) of the largest (or the smallest) of them. For example: $$ \text{if } Z=\max_i{X_i}\text{ , then }P(Z\leq t)=P(\cap_i\{X_i\leq t\})=P(X_i\leq t)^n $$ However, I am interested in the probability function (or more specifically, the expectation) of the k'th largest variable, call it $Z_k$ (so $Z$ from above is just $Z_1$, and the smallest is $Z_n$).
The first step I thought about was to note that $Z_k$ satisfies 2 conditions: it is the maximum of the set $Z_k,Z_{k+1},...,Z_n$, and the minimum of the set $Z_1,Z_2,...,Z_k$. But I couldn't think of a way to use it. If it simplifies things, it can be assumed that the $X_i$s are non-negative.