Chap 9 Multivariate Distributions Ghahramani 3rd edition 2019/4/17

Outline 9.1 Joint distribution of n>2 random variables 9.2 Order statistics 9.3 Multinomial distributions

Skip 9.1 Joint distributions of n>2 random variables 9.3 Multinomial Distributions

9.2 Order Statistics Def Let {X1, X2, …, Xn} be an independent set of identically distributed continuous random variables with the common density and distribution functions f and F. Let X(1) be the smallest value in {X1, X2, …, Xn} , X(2) be the second smallest value in {X1, X2, …, Xn} , X(3) be the third smallest, and, in general, X(k) (1<=k<=n) be the kth smallest value in {X1, X2, …, Xn} . Then X(k) is called the kth order statistic.

Ex 9.6 Suppose that customers arrive at a warehouse from n different locations. Let Xi, 1<=i<=n, be the time until the arrival of the next customer from location i; then X(1) is the arrival time of the next customer to the warehouse.

Ex 9.7 Suppose that a machine consists of n components with the lifetimes X1, X2, …, Xn, where Xi’s are i.i.d.. Suppose that the machine remains operative unless k or more of its components fail. Then X(k), the kth order statistic of {X1, X2, …, Xn}, is the time when the machine fails. Also, X(1) is the failure time of the first component.

Ex 9.8 Let X1, X2, …, Xn be a random sample of size n from a population with continuous distribution F. Then the following important statistical concepts are expressed in terms of order statistics: (i) The sample ranges is X(n) - X(1). (ii) The sample midrange is [X(n) + X(1)]/2. (iii) The sample median is

Thm 9. 5 Let {X(1), X(2), …, X(n)} be the order statistics of i. i. d Thm 9.5 Let {X(1), X(2), …, X(n)} be the order statistics of i.i.d. continuous r.v.’s with the common density and distribution functions f and F. Then Fk and fk, the prob. distribution and prob. density functions of X(k), respectively, are given by

Remark 9.2 (Derive F1, f1, Fn, fn directly)

Ex 9.9 Let X1, X2, …, X2n+1 be 2n+1 i.i.d. random numbers from (0,1). Find the prob. density function of X(n+1). Sol: