Chap 9 Multivariate Distributions Ghahramani 3rd edition

1
Chap 9 Multivariate DistributionsGhahramani 3rd
edition
2
Outline

• 9.1 Joint distribution of ngt2 random variables
• 9.2 Order statistics
• 9.3 Multinomial distributions

3

• Skip
• 9.1 Joint distributions of ngt2 random variables
• 9.3 Multinomial Distributions

4
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) (1ltkltn) be the kth smallest value in X1,
  X2, , Xn . Then X(k) is called the kth order
  statistic.

5

• Ex 9.6 Suppose that customers arrive at a
  warehouse from n different locations. Let Xi,
  1ltiltn, 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.

6

• Ex 9.7 Suppose that a machine consists of n
  components with the lifetimes X1, X2, , Xn,
  where Xis 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.

7

• 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

8

• 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

9

10

11

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

12

13

• Ex 9.9 Let X1, X2, , X2n1 be 2n1 i.i.d.
  random numbers from (0,1). Find the prob.
  density function of X(n1).
• Sol