LESSON 5.2 · ABOUT 20 MINUTES
Order statistics
Study the smallest, largest, and ranked observations.
By the end of this lesson
- Find distributions of minima and maxima.
- Use ranks and the joint density of extreme observations.
Minima and maxima
For independent observations with CDFs Fᵢ, P(max≤x)=∏Fᵢ(x); P(min>x)=∏(1−Fᵢ(x)). For identically distributed samples these become F(x)ⁿ and (1−F(x))ⁿ.
The kth observation
For n independent observations with common continuous CDF F, P(X_(k)≤x)=Σ from j=k to n of C(n,j)F(x)^j(1−F(x))^(n−j). Count how many observations fall below x. The kth order statistic from Uniform(0,1) is Beta(k,n+1−k).
Joint order statistics
For an i.i.d. continuous sample with density f, the joint density of minimum u and maximum v is n(n−1)[F(v)−F(u)]^(n−2)f(u)f(v), for u<v. One observation occupies each endpoint and the others lie in between. More generally, the joint density of ranks r<s is n!/[(r−1)!(s−r−1)!(n−s)!] × F(u)^(r−1)[F(v)−F(u)]^(s−r−1)[1−F(v)]^(n−s)f(u)f(v).
P(max≤x)=F(x)ⁿ; P(min>x)=[1−F(x)]ⁿ (i.i.d. sample)
Worked example
Three independent Uniform(0,1) observations are drawn. Find P(min>0.2 and max≤0.8).
Every observation must fall in (0.2,0.8], an interval of probability 0.6. Independence gives 0.6³=0.216.