Order Statistics

Find the distribution of the minimum, maximum, and general order statistics of independent random variables.

1. The idea

Given iid random variables $X_{1}, \dots, X_{n}$ , the order statistics $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$ are the sorted values. The maximum $X_{(n)}$ and minimum $X_{(1)}$ are the most commonly tested cases.

The CDF of the maximum is easy because "max $\leq x$ " happens exactly when every variable is $\leq x$ : $P (X_{(n)} \leq x) = F (x)^{n}$ . Similarly, the minimum's survival function factors: $P (X_{(1)} > x) = (1 - F (x))^{n}$ , since the minimum exceeds $x$ only if all variables do.

For continuous distributions, differentiate the CDF to get the density of an order statistic, or use the general formula involving a binomial coefficient counting arrangements. Interview problems most often need only max/min or expected values for specific distributions like the uniform.

2. Why it matters

Order statistics appear whenever a quant question involves 'the best of n draws' or 'time until the last event' — auctions, extreme values, record times, and worst-case risk scenarios all reduce to max/min reasoning.

3. Core mathematics

CDF of the maximum

P (X_{(n)} \leq x) = [F (x)]^{n}

Survival function of the minimum

P (X_{(1)} > x) = [1 - F (x)]^{n}

Expected max of n iid Uniform(0,1)

E [X_{(n)}] = \frac{n}{n + 1}

Expected min of n iid Uniform(0,1)

E [X_{(1)}] = \frac{1}{n + 1}

4. How to recognize it

Think of this when…

The problem asks about the largest, smallest, or median of several iid random variables.
"Time until the last of n events" or "time until the first of n events" phrasing.
Auction or record-value framing: highest bid, best offer, worst outcome among n trials.
You need $E [max]$ or $E [min]$ for a named distribution like uniform or exponential.

5. Worked example

Worked example

$X_{1}, \dots, X_{n}$ are iid Uniform(0,1). Find $E [X_{(n)}]$ , the expected maximum.

1
The CDF of the maximum is $F_{(n)} (x) = P (X_{(n)} \leq x) = x^{n}$ for $x \in [0, 1]$ , since all $n$ variables must be $\leq x$ .
2
Differentiate to get the density: $f_{(n)} (x) = n x^{n - 1}$ .
3
Compute the expectation by integrating.
$E [X_{(n)}] = \int_{0}^{1} x \cdot n x^{n - 1} d x = n \int_{0}^{1} x^{n} d x = \frac{n}{n + 1}$
Answer: $E [X_{(n)}] = \frac{n}{n + 1}$

6. Common mistakes

Confusing the CDF of the max, $F (x)^{n}$ , with the CDF of the min, which requires working with the survival function $1 - F (x)$ instead.
Forgetting that order statistic formulas assume independence (and usually identical distribution) among the variables.
Trying to compute $E [max]$ via linearity of expectation directly on the maxima, which does not work since max is not linear.
Mixing up $n / (n + 1)$ and $1/ (n + 1)$ for the expected max vs. min of uniforms.

7. Quick check

For 3 iid Uniform(0,1) variables, what is $E [min]$ ?

For iid random variables with CDF $F$ , $P (X_{(1)} > x)$ equals:

8. Practice

29 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Order Statistics

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