In this problem, we would like to find the PDFs of 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 Our goal here is to show that One way to do this is to differentiate the CDF (found in Pr...

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In this problem, we would like to find the PDFs of order statistics. Let X 1 ,,X

Question:

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

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 fx) (x) P(x X (i) x + 8)d. Note that the event {x X(i) x +8} occurs if i - 1 of the X;'s are less than

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 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)).

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