730 Lecture 4 1022020 730 Lecture 4 1

730: Lecture 4 10/2/2020 730 Lecture 4 1

Today’s menu… Examples on • Order Statistics • Change of variable 10/2/2020 730 Lecture 4 2

Example 1 At least k x Number of events occurring (Y) is Bin(n, F(x)) 10/2/2020 730 Lecture 4 3

Example 1 (cont) Take f uniform[0, 1]. Now density of X(k) is ie Beta (k, n-k+1) If n=2 m+1, median is X(m+1), has Beta(m+1, m+1) distribution 10/2/2020 730 Lecture 4 4

Example 1 (cont) § Mean of Beta(a, b) is a/(a+b) § Variance of Beta(a, b) is ab/((a+b)2(a+b+1) § Thus mean and variance of the median are 0. 5 and 1/(4(2 m+3)) § Compare with mean, mean is 0. 5 but variance is 1/(12(2 m+1)) (variance of U[0, 1] is 1/12) 10/2/2020 730 Lecture 4 5

Example 2 The event {x<X(1) , X(n) £y} occurs iff all the events {x< Xi £y} occur. Thus P(x<X(1) , X(n) £ y) =PP(x< Xi £ y ) = [F(y)-F(x)]n 10/2/2020 730 Lecture 4 6

Example 2 (cont) Use formula P(CAÇB) = P(B) – P(AÇB) Get P[X(1) £x, X(n) £y] = P[X(n) £y] - P[x<X(1), X(n) £y] =F(y)n - (F(y)-F(x))n 10/2/2020 730 Lecture 4 7

Example 2 (cont) In uniform case, we get joint df G(x, y) = yn – (y-x)n, y³x Joint density is g(x, y)=n(n-1)(y-x)n-2, y³x Because: 10/2/2020 730 Lecture 4 8

Example 3 Recall change of variable formula: 10/2/2020 730 Lecture 4 9

Example 3 (cont) Apply to sample range: r=g 1(x(1), x(n) )= x(n) - x(1) t=x(1) J=-1 10/2/2020 730 Lecture 4 10

Example 3 (Cont) Marginal density of R is 10/2/2020 730 Lecture 4 11