In this problem, we would like to find the PDFs of order statistics. Let X₁,…,X_n be a random sample from a continuous distribution with CDF F_X(x) and PDF f_X(x). Define X₍₁₎,…,X_(n) as the order statistics. Our goal here is to show that

One way to do this is to differentiate the CDF (found in Problem 9). However, here, we would like to derive the PDF directly. Let f_X(i) (x) be the PDF of X_(i). By definition of the PDF, for small δ, we can write 

Problem 9

In this prob lem, we would like to find the CDFs of the order statistics. Let X₁,…,X_n be a random sample from a continuous distribution with CDF F_X(x) and PDF f_X(x).

Define X₍₁₎,…,X_(n) as the order statistics and show that

Fix x ∈ R. Let Y be a random variable that counts the number of X_j's ≤ x. Define {X_j ≤ x} as a "success" and {X_j > x} as a "failure," and show that Y ∼ Binomial(n, p = FX(x)).

fx(i)(x) = = n! fx(x) [Fx(x)] [1 - Fx(z)]". (i-1)! (ni)! *