1 / 11

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.

johnstoncynthia
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related