Order Statistics: Min & Max

By the end of this page you'll derive the CDF and density of the largest and smallest of n i.i.d. draws, straight from F(x), using an "all of them" or "none of them" argument.

Predict first: as you slide n up, which way does the min's density pile up — toward 0 or toward 1? And the max's? Now slide n and draw n i.i.d. Uniform(0,1) points to check. The min (green) and max (orange) are highlighted on the line; the density and CDF panels update live. Hover any point or curve for exact values.

n = 5 No draws yet.

Sampled draws — n i.i.d. Uniform(0,1), min & max highlighted

Densities of the min and max

f_min(x) = n(1−x)^(n−1) f_max(x) = n·x^(n−1)

CDFs of the max and min

P(max ≤ x) = xⁿ P(min > x) = (1−x)ⁿ

For n i.i.d. continuous variables, the minimum and maximum — two of the order statistics of the sample — have CDFs derived directly from \(F(x)\): \(P(\max \le x) = F(x)^n\) and \(P(\min > x) = (1-F(x))^n\), with general order-statistic densities following by differentiation.

Formal

Max: the max is \(\le x\) iff every \(X_i \le x\) — by independence, probabilities multiply:

\(F_{\max}(x) = P(X_1 \le x, \ldots, X_n \le x) = F(x)^n \;\Rightarrow\; f_{\max}(x) = n \cdot F(x)^{n-1} f(x)\)

Min: easier from the complement — the min is \(> x\) iff every \(X_i > x\):

\(P(\)\(X_{(1)}\)\( > x) = (1-F(x))^n \;\Rightarrow\; F_{\min}(x) = 1 - (1-F(x))^n \;\Rightarrow\; f_{\min}(x) = n(1-F(x))^{n-1} f(x)\)

General k-th order statistic: \(f_{(k)}(x) = \dfrac{n!}{(k-1)!(n-k)!} F(x)^{k-1} (1-F(x))^{n-k} f(x)\)

Applied (actuarial)

A reinsurance treaty pays only the largest of n independent claims in a layer (e.g. largest-claim reinsurance), or a warranty triggers on the first (minimum time-to-failure) of n components. Pricing either product requires the max or min distribution derived above, not the individual claim/lifetime distribution.

Worked example

\(X_1, X_2, X_3\) i.i.d. Uniform(0,1), \(F(x) = x\). \(f_{\max}(x) = 3x^2\), \(E[\max] = \int_0^1 x \cdot 3x^2\,dx = 3/4\). \(f_{\min}(x) = 3(1-x)^2\), \(E[\min] = \int_0^1 x \cdot 3(1-x)^2\,dx = \) 1/4 (symmetric with max about 1/2).

Now you try — faded example

\(X_1, \ldots, X_4\) i.i.d. Uniform(0,1). Find \(E[\min]\).

Step 1 — \(F_{\min}(x) = 1-(1-x)^4 \;\Rightarrow\; f_{\min}(x) = 4(1-x)^3\).

Step 2 — \(E[\min] = \int_0^1 x \cdot 4(1-x)^3\,dx\).

Step 3 — finish it: evaluate the integral =

Reveal the answer

1/5. In general, for n i.i.d. Uniform(0,1) draws, \(E[\min] = 1/(n+1)\) — with n = 4, that's 1/5, matching the integral.

More info — order statistics as a special case of a joint density

Here's another way to see the max and min: they're just the two edges of the full joint density of every order statistic together, \(f_{(1),\ldots,(n)}(x_1,\ldots,x_n) = n!\,f(x_1)\cdots f(x_n)\) for \(x_1 < \cdots < x_n\) (the \(n!\) counts the orderings of the original sample that land in this one sorted arrangement). Integrating that joint density down to just the first and last coordinates recovers the min and max formulas above — the Harvard Stat 110 lecture and the "Joint PDF of the Min and Max" video under Dive deeper below both walk through that derivation.

Check your understanding

Question 1 of 4

For n i.i.d. continuous variables with CDF F(x), what is P(max(X₁,...,Xₙ) ≤ x)?

Question 2 of 4

For n=3 i.i.d. Uniform(0,1) variables, what is the density of the minimum, f_min(x)?

Question 3 of 4

For n = 1 (a single i.i.d. draw), what are F_max(x) and F_min(x)?

Question 4 of 4

Four independent component lifetimes are i.i.d. Exponential(λ=2), so F(t) = 1 - e^(-2t). What is P(the earliest failure happens after t=1), i.e. P(min > 1)?

Recap

Max: \(F_{\max}(x) = F(x)^n\) — the max is \(\le x\) iff every draw is, and independence multiplies the probabilities.
Min: \(F_{\min}(x) = 1-(1-F(x))^n\) — easiest from the complement: the min is \(>x\) iff every draw is.
Differentiate to get densities: \(f_{\max}(x) = nF(x)^{n-1}f(x)\), \(f_{\min}(x) = n(1-F(x))^{n-1}f(x)\).
General \(k\)-th order statistic: \(f_{(k)}(x) = \dfrac{n!}{(k-1)!(n-k)!} F(x)^{k-1}(1-F(x))^{n-k}f(x)\).
As n grows, the max's density piles up near the top of the support and the min's near the bottom — watch it in the interactive above.

Dive deeper

Harvard Stat 110 — Lecture 25: Order Statistics The theory behind the min and max distributions.
Joint PDF of the Min and Max of a Random Sample Derive the joint density of the min and the max.
Order Statistics: Derivation of PDF and Example Work a general order-statistic density example.

Sources

Order Statistics — Min, Max, and General Rank Distributions