Order Statistics

If you take a sample and sort it from smallest to largest, the resulting sorted values are the order statistics. The smallest value is the first order statistic and the largest value is the last order statistic.

This is a natural exam topic because minima and maxima are joint events in disguise. A maximum being below a threshold means every observation is below that threshold. A minimum being above a threshold means every observation is above that threshold.

Suppose X1, X2, and X3 are iid Uniform(0,1). The probability that the maximum is at most 0.8 is the probability that all three observations are at most 0.8.

So P(X(3) <= 0.8) = 0.8^3 = 0.512. That is the pattern to remember: convert the order-statistics statement into an all-of-them or at-least-one-of-them event, then use independence cleanly.

The syllabus explicitly includes the joint distribution of order statistics for independent random variables. In practice, the entry point for many candidates is still the minimum and maximum logic because that is where the structure becomes intuitive.

Even when a question looks unfamiliar, it usually becomes easier once you translate the statement about the sorted sample into a probability statement about the original unsorted variables.

The usual mistake is flipping the complement incorrectly. For maxima, candidates sometimes compute the probability that at least one observation is below the threshold instead of all observations being below it. For minima, the same kind of error happens in reverse.

Order statistics sit behind quantiles, sample extremes, tail modeling, and reliability questions. They are also the conceptual doorway into more advanced extreme-value thinking later on.