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Exploring Order Statistics in Probability | Understand Mean, Variance, and Density Functions

Learn about order statistics in probability theory through examples on change of variables, calculations, and distributions. Discover the mean, variance, and density functions of different statistical events.

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Exploring Order Statistics in Probability | Understand Mean, Variance, and Density Functions

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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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