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Lecture

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School
Department
Statistics
Course
STAT 330
Professor
Christine Dupont
Semester
Fall

Description
A Review of ’Convergence in Distribution to a Degenerate r.v.’ Examples, with a focus on discontinuity points e.g. 1) Y = max(X ;:::;X ) where X ;:::;X ▯ Unif(0;1). Find the limiting distribution of n 1 n 1 n Yn. Sol. Step 1: Recall 8 <0 ; y < 0 P(Y ▯ y) = n n > y ; 0 ▯ y < 1 : 1 ; y ▯ 1 Step 2: ( 0 ; y < 1 lim P(Yn▯ y) = n!1 1 ; y ▯ 1 n Note the point of discontinuity at y = 1: Checn P(Y ▯ 1) = 1 = 1 ! 1 as n ! 1. iid e.g. 2) n = min(X 1:::;X n where X 1:::;X n ▯ Unif(0;1). Find the limiting distribution of Y . n Sol. Step 1: Recall 8 >0 ; y < 0 < P(Yn▯ y) = 1 ▯ [1 ▯ y] ; 0 ▯ y < 1 : 1 ; y ▯ 1 Step 2: ( 0 ; y ▯ 0 n!1m P(Yn▯ y) = 1 ; y > 0 Note the point of discontinuity at y = 0: Check P(Y ▯ 0) = 1 ▯ [1 ▯ 0] = 1 ▯ 1 = n 1 ▯ 1 = 0 ! 0 as n ! 1. iid ▯(x▯▯) e.g. 3) (Example 5.2.5 revisitnd) Y = ma1(X ;::n;X ) wher1 X ;:::nX ▯ F(x) = 1▯e ; x > ▯ and F(x) = 0; x ▯ ▯. Find the limiting distribution of Y . n Sol. Step 1: Recall ( 1 ▯ en(y▯▯) ; y > ▯ P(Yn▯ y) = 0 ; y ▯ ▯ 1
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