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730: Lecture 4

730: Lecture 4. Examples. Today’s menu…. Examples on Order Statistics Change of variable. At least k. x. Example 1. Number of events occurring (Y) is Bin(n,F(x)). Example 1 (cont). Take f uniform[0,1]. Now density of X (k) is. ie Beta (k,n-k+1).

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730: Lecture 4

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  1. 730: Lecture 4 Examples 730 Lecture 4

  2. Today’s menu… Examples on Order Statistics Change of variable 730 Lecture 4

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

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

  5. 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(2m+3)) • Compare with mean, mean is 0.5 but variance is 1/(12(2m+1)) (variance of U[0,1] is 1/12) 730 Lecture 4

  6. 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 730 Lecture 4

  7. 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 730 Lecture 4

  8. Because: 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 730 Lecture 4

  9. Example 3 Recall change of variable formula: 730 Lecture 4

  10. Example3 (cont) Apply to sample range: r=g1(x(1), x(n) )= x(n) - x(1) t=x(1) J=-1 730 Lecture 4

  11. Example 3 (Cont) Marginal density of R is 730 Lecture 4

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