Order Statistics

Probability · Axiom Academy

Understanding X₍₁₎ ≤ X₍₂₎ ≤ ... ≤ X₍ₙ₎, distributions of min/max, and PDF of kth order statistic 1. Minimum and Maximum Distributions For independent and identically distributed (i.i.d.) random variables X₁, ..., Xₙ with CDF F(x), the minimum X₍₁₎ and maximum X₍ₙ₎ have simple CDFs: The minimum is less than x only if all observations exceed x (complement). The maximum is less than x only if all observations are less than x. 2. Distribution of kth Order Statistic The PDF of the kth order statistic X₍ₖ₎ combines three events: (k-1) values below x, one value at x, and (n-k) values above x. Using combinatorics: This formula accounts for all possible arrangements where exactly k-1 observations fall below x, exactly 1 observation is near x (contributing f(x)), and n-k observations are above x. 3. Median and Sample Quantiles The sample median is the middle order statistic. For n = 2m+1 (odd), it's X₍ₘ₊₁₎. For general quantiles, the pth quantile corresponds to X₍ₖ₎ where k ≈ np: As n → ∞, order statistics concentrate around population quantiles, making them consistent estimators.

This is the written version of the interactive lesson above. See the full Probability course.