In this problem, we would like to find the CDFs of the order statistics Let X 1 ,,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)) n Fxs) (x) (7) Fx(2) 1 Fx(x) (i) k k i

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Question: In this problem, we would like to find the CDFs of the order statistics. Let X 1 ,,X n be a random sample from a

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

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)).

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

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