In this problem, 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)).

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n Fxs) (x) = Σ (7) [Fx(2)] * [1 − Fx(x)]*-*. (i) k k=i