Order Statistics and Extremes

"What's the expected largest of five draws?" — questions about maxima, minima, and ranks form their own genre, and they all yield to one opening move: work with the CDF of the extreme, not its density.

The CDF trick

For iid draws $X_1, \ldots, X_n$ with CDF $F$:

$$P(\max \le x) = F(x)^n, \qquad P(\min > x) = (1 - F(x))^n$$

Everything else follows. The max-of-two-dice question from the practice bank was this with $F(k) = k/6$; the general recipe is identical for any distribution.

Uniforms: the clean benchmark

For $n$ iid Uniform$(0,1)$ draws, the $k$-th smallest has expectation

$$E[U_{(k)}] = \frac{k}{n+1}$$

— the $n$ points split the interval into $n+1$ statistically identical gaps, each of expected length $\frac{1}{n+1}$. So the expected max of 4 uniforms is $4/5$; the expected minimum is $1/(n+1)$. This spacings symmetry is the same argument that placed the first ace at position $(52+1)/(4+1)$ — one idea, many costumes. Exponentials have their own beautiful version: the minimum of exponentials is exponential with summed rates (the first of many alarms rings at the combined rate), and $E[\max]$ of $n$ unit exponentials is $H_n$ — the harmonic number from the series lesson.

Records: how often is a new maximum set?

Watching iid draws arrive, the $k$-th is a record (beats all before it) with probability $1/k$ — by symmetry, each of the first $k$ is equally likely to be the largest. So expected records among $n$ draws is $H_n \approx \ln n + 0.577$: about 5 records in 100 draws, 7 in 1,000. Records are logarithmically rare — the crisp probabilistic answer to why "all-time highs" cluster early in any series and why a new "best backtest ever" among thousands tried means very little (selection over $n$ trials inflates the best by roughly the max-of-$n$ machinery above).

Extremes and quoting

Two desk-shaped consequences:

• Expected max grows slowly. For normals, $E[\max \text{ of } n] \approx \sigma\sqrt{2 \ln n}$: the worst of 252 trading days is only ~3.3σ if returns were normal — observed worst days are far worse, which is the fat-tails lesson quantified through extremes. Extreme value theory (Gumbel/Fréchet limits) formalizes this; for interviews, the $\sqrt{2\ln n}$ scaling plus the caveat is enough.
• Winner's-curse sizing: the highest of $n$ noisy estimates overshoots the truth by roughly the expected max of the noise — the game-theory lesson's auction shading, now with a formula behind it.

The interview version

"Five uniform random numbers on [0,1]. Expected value of the largest? The second largest? P(largest > 0.9)?" — $5/6$, $4/6$, and $1 - 0.9^5 \approx 41\%$. Three answers, two tools (spacings, CDF-power), ten seconds each. Then the standard escalation: "you see the values one at a time and must stop on one — how does the max help you?" — and you're back at the secretary problem, which is order statistics plus optimal stopping. The genre is small; these tools cover it.