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###### 日期：2020-02-07 10:10

Queen’s University

Department of Mathematics and Statistics

MTHE/STAT 353

Homework 3 Due February 6, 2020

? For each question, your solution should start on a fresh page. You can write your

solution using one of the following three formats:

(1) Start your solution in the space provided right after the problem statement,

and use your own paper if you need extra pages.

(2) Write your whole solution using your own paper, and make sure to number

(3) Write your solution using document creation software (e.g., Word or LaTeX).

? Write your name and student number on the first page of each solution.

? For each question, photograph or scan each page of your solution (unless your solution

has been typed up and is already in electronic format), and combine the

separate pages into a single file. Then upload each file (one for each question), into

the appropriate box in Crowdmark.

MTHE/STAT 353 -- Homework 3, 2020 2

Student Number Name

1. Let X1, . . . , Xn be independent and identically distributed continuous random variables

and let X(1), . . . , X(n) denote their order statistics.

(a) Find P(Xn = X(k)) for k = 1, . . . , n. Hint: Note that all orderings of X1, . . . , Xn

are equally likely.

(b) Show that (Xn, X(n)) does not have a joint pdf.

MTHE/STAT 353 -- Homework 3, 2020 3

Student Number Name

2. Let X1, . . . , Xn be independent exponential random variables with parameter λ, and let

X(1), . . . , X(n) be their order statistics. Show that

Y1 = nX(1), Yr = (n + 1 ? r)(X(r) ? X(r?1)), r = 2, . . . , n

are also independent and have the same joint distribution as X1, . . . , Xn. Hint: You may

use the fact that the determinant of a lower triangular matrix (a square matrix whose

entries above the main diagonal are all zero) is the product of the diagonal entries.

MTHE/STAT 353 -- Homework 3, 2020 4

Student Number Name

3. Let X1, X2, X3 be independent, identically distributed continuous random variables. Find

the probability that the second largest value (i.e., the median) is closer to the smallest

value than to the largest value, when the common distribution of the Xi

is

(a) the Uniform(0, 1) distribution (a symmetry argument should suffice here);

(b) the Exponential(λ) distribution.

MTHE/STAT 353 -- Homework 3, 2020 5

Student Number Name

4. Let X1, . . . , Xn be mutually independent Uniform(0,1) random variables. Find the probability

that the interval (min(X1, . . . , Xn), max(X1, . . . , Xn)) contains the value 1/2 and

find the smallest n such that this probability is at least 0.95.

MTHE/STAT 353 -- Homework 3, 2020 6

Student Number Name

5. Let X1, X2, . . . be a sequence of independent random variables with the exponential distribution

with mean 1, and let X(n) = max(X1, . . . , Xn). For x > 0, show that

limn→∞P(X(n) ? ln n ≤ x) = exp(?e?x).