MTHE/STAT 353 Winter 2014
Homework Assignment 9
Assignment 9 due Thursday, April 3
1. Let X1 , X2 , . . . be a sequence of random variables and let c be a constant. Show that
if, as n , Xn c in distribution then Xn c in probability.
2. (a) Let X1 , X2 , .
MTHE/STAT 353 Winter 2014
Homework Assignment 5
Assignment 5 due Monday, March 3
1. Let (X1 , . . . , Xk ) have a multinomial distribution with parameters n and p1 , . . . , pk .
For i = 1, . . . , k, nd Cov(Xi , Xj ) and (Xi , Xj ). Hint: Write Xi = Xi1
MTHE/STAT 353 Winter 2014
Homework Assignment 2
Assignment 2 due Monday, Jan. 27
1. A product set in Rn is a set S of the form S = S1 S2 . . . Sn , where Si R for
i = 1, . . . , n, and the set S1 . . . Sn is dened as
S1 . . . Sn = cfw_(x1 , . . . xn ) Rn
MTHE/STAT 353 Winter 2014
Homework Assignment 1
Assignment 1 due Thursday, Jan. 16
1. An urn contains 12 balls: 3 red balls, 3 blue balls, 3 green balls, and 3 magenta balls.
Nine of the balls are selected at random and removed from the urn.
(a) Let X den
ERRATA
FOR
FUNDAMENTALS
OF PROBABILITY
WITH
STOCHASTIC PROCESSES
Third Edition
1.
On Page 159, cut the last sentence of Exercise 14, Count the letter Y as a
consonant, and paste it to the end of Exercise 13.
2.
On Page 223, lines 17 and 20, change x to i
MTHE/STAT 353 Winter 2014
Homework Assignment 3
Assignment 3 due Monday, Feb. 3
1. Let X1 , . . . , Xn be independent exponential random variables with parameter , and
let X(1) , . . . , X(n) be their order statistics. Show that
Y1 = nX(1) ,
Yr = (n + 1 r
STAT 353 Solutions: Assignment 7
Winter, 2016 (Total 30 marks)
Problem 1 (From Sheet.) (6 marks)
(a) (3 marks) Let X denote the number of flips required until at least one head and one
tail have been flipped and let Y indicate the outcome of the first fli
STAT 353 Solutions: Assignment 4
Winter, 2016 (Total 28 marks)
Problem 1 (From Sheet.) (5 marks) For a given n, the probability we want to compute is
P (X(1) < 1/2, X(n) > 1/2). The joint pdf of (X(1) , X(n) ) is given by
(
n(n 1)(xn x1 )n2 for 0 < x1 < x
MTHE/STAT 353 Solutions: Assignment 1
Winter, 2016 (Total 26 marks)
Problem 1 (From Sheet.) (6 marks)
(a) (3 marks) After 9 draws, the total number of red, blue and green balls must be at least
6, so the possible values of (X, Y, Z) are those in the set
S
STAT 353 Solutions: Assignment 8
Winter, 2016 (Total 32 marks)
Problem 1 (From Sheet.) (4 marks) Let MX (t) denote the moment generating function of
X. Following the hint, writing MX (t) in a Taylor series expansion, we have
tX
MX (t) = E[e ] = E
hX
(tX)n
STAT 353 Solutions: Assignment 3
Winter, 2016 (Total 27 marks)
Problem 1 (From Sheet.) (6 marks)
(a) (3 marks) From the binomial theorem
k
k )
1
1
1
1
1
1
+
+ 1
2
n
n
n
n
( k
km m X
km
m )
k
1 X k
1
1
k
1
1
=
1
+
1
2 m=0 m
n
n
m
n
n
m=0
( k
)
km m
STAT 353 Solutions: Assignment 6
Winter, 2016 (Total 25 marks)
Problem 1 (From Sheet.) (4 marks) On problem 5 of homework 4 it was computed that
E[Xi Xj ] = n(n 1)pi pj . Therefore, to compute Cov(Xi , Xj ) and (Xi , Xj ), we further
need to compute E[Xi
STAT 353 Solutions: Assignment 5
Winter, 2016 (Total 25 marks)
Problem 1 (From Sheet.) (4 marks)
(a) (2 marks) We can define three random variables, say Y1 , Y2 and Y3 , where Y1 is the
number of Xs that are less than .2, Y2 is the number of Xs that are g
MTHE/STAT 353 Solutions: Assignment 2
Winter, 2016 (Total 30 marks)
Problem 1 (From Sheet.) (5 marks) First, the marginal distribution of each Xi is given by
P (Xi = 1) = P (both endpoints of edge i have the same value)
6
X
=
P (both endpoints of edge i h
STAT/MTHE 353 - Probability II
Winter 2014
Instructor:
Glen Takahara - Jeffery Hall 407
Phone: 533-2430, Email: takahara@mast.queensu.ca
Course Web Site: http:/www.mast.queensu.ca/ stat353
All assignments and important announcements will be posted here.
L
MTHE/STAT 353 Winter 2014
Homework Assignment 4
Assignment 4 due Monday, February 10
1. For n = 0, 1, 2, 3, . . ., show that
(n + 1/2) =
(2n)!
.
4n n!
2. Let X and Y be independent random variables, each with a normal distribution with
mean 0 and standard
MTHE/STAT 353 Winter 2014
Homework Assignment 6
Assignment 6 due Monday, March 10
1. Let X1 and X2 be random variables and Y a random vector. Show the following
generalization of the conditional variance formula:
Cov(X1 , X2 ) = E[Cov(X1 , X2 ) Y ] + Cov(
Student Number
Queens University
Department of Mathematics and Statistics
MTHE/STAT 353
Midterm Examination February 13, 2014
Total points = 30. Duration = 58 minutes.
This is a closed book exam.
One 8.5 by 11 inch sheet of notes, written on both sides
Student Number
Queens University
Department of Mathematics and Statistics
STAT 353
Final Examination April 24, 2010
Instructor: G. Takahara
Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam
Student Number
Queens University
Department of Mathematics and Statistics
STAT/MTHE 353
Final Examination April 21, 2012
Instructor: T. Linder
PLEASE NOTE: Proctors are unable to respond to queries about the interpretation
of exam questions. Do your best
Student Number
Queens University
Department of Mathematics and Statistics
MTHE/STAT 353
Final Examination April 13, 2013
Instructor: G. Takahara
Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer
Student Number
Queens University
Department of Mathematics and Statistics
STAT 353
Final Examination April 9, 2009
Instructor: G. Takahara
Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam q
Student Number
Queens University
Department of Mathematics and Statistics
STAT 353
Final Examination April 21, 2011
Instructor: G. Takahara
Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam
Student Number
Queens University
Department of Mathematics and Statistics
STAT 353
Midterm Examination February 18, 2011
Total points = 30. Duration = 58 minutes.
This is a closed book exam.
One 8.5 by 11 inch sheet of notes, written on both sides, is
Student Number
Queens University
Department of Mathematics and Statistics
STAT 353
Midterm Examination February 12, 2009
Total points = 30. Duration = 58 minutes.
This is a closed book exam.
One 8.5 by 11 inch sheet of notes, written on both sides, is
Student Number
Queens University
Department of Mathematics and Statistics
STAT 353
Midterm Examination February 18, 2010
Total points = 30. Duration = 58 minutes.
This is a closed book exam.
One 8.5 by 11 inch sheet of notes, written on both sides, is
Student Number
Queens University
Department of Mathematics and Statistics
STAT/MTHE 353
Midterm Examination February 15, 2013
Total points = 30. Duration = 58 minutes.
This is a closed book exam.
One 8.5 by 11 inch sheet of notes, written on both sides
Student Number
Queens University
Department of Mathematics and Statistics
STAT/MTHE 353
Midterm Examination February 16, 2012
Total points = 30. Duration = 58 minutes.
This is a closed book exam.
One 8.5 by 11 inch sheet of notes, written on both sides
MTHE/STAT 353 Winter 2014
Homework Assignment 8
Assignment 8 due Monday, March 24
1. Let X have a Poisson distribution with mean and let k be a positive integer. Find
the upper bound to P (X k) given by Markovs inequality, Chebyshevs inequality,
and Chern
MTHE/STAT 353 Winter 2014
Homework Assignment 7
Assignment 7 due Monday, March 17
1. Let X1 , X2 , . . . be independent and identically distributed with common moment generating function mX (t) and let N be a positive integer valued random variable, indep