1. Probability theory 2010 Order statistics  Distribution of order variables (and extremes)  Joint distribution of order variables (and extremes)

2. Probability theory 2010 Order statistics Let X 1, …, X n be a (random) sample and set X (k) = the kth smallest of X 1, …, X n Then the ordered sample (X (1), X (2), …, X (n) ) is called the order statistic of (X 1, …, X n ) and X (k) the kth order variable

3. Probability theory 2010 Order variables - examples Example 1: Let X 1, …, X n be U(0,1) random numbers. Find the probability that max(X 1, …, X n ) > 1 – 1/n Example 2: Let X 1, …, X 100 be a simple random sample from a (finite) population with median m. Find the probability that X (40) > m.

4. Probability theory 2010 Distribution of the extreme order variables

5. Probability theory 2010 The beta distribution For integer-valued r and s, the beta distribution represents the rth highest of a sample of r+s-1 independent random variables uniformly distributed on (0,1)  =r  =s

6. Probability theory 2010 The gamma function

7. Probability theory 2010 Distribution of arbitrary order variables

8. Probability theory 2010 A useful identity Can be proven by backward induction

9. Probability theory 2010 Distribution of arbitrary order variables

10. Probability theory 2010 Distribution of arbitrary order variables from a U(0,1) distribution

11. Probability theory 2010 Joint distribution of the extreme order variables

12. Functions of random variables Let X have an arbitrary continuous distribution, and suppose that g is a (differentiable) strictly increasing function. Set Then and

13. Linear functions of random vectors Let (X 1, X 2 ) have a uniform distribution on D = {(x, y); 0 < x <1, 0 < y <1} Set Then.

14. Functions of random vectors Let (X 1, X 2 ) have an arbitrary continuous distribution, and suppose that g is a (differentiable) one-to-one transformation. Set Then where h is the inverse of g. Proof: Use the variable transformation theorem

15. Probability theory 2010 Density of the range Consider the bivariate injection Then and

16. Probability theory 2010 Density of the range

17. Probability theory 2010 The range of a sample from an exponential distribution with mean one Probabilistic interpretation of the last equation?

18. Probability theory 2010 Joint distribution of the order statistic Consider the mapping (X 1, …, X n )  (X (1), …, X (n) ) or. where P is a permutation matrix

19. Probability theory 2010 Joint density of the order statistic

20. Probability theory 2010 Exercises: Chapter IV 4.2, 4.7, 4.16, 4.19, 4.21