UNIVERSITY OF
WAT E R LOO Final Examination
Term: Spring Year: 2013
Student Name: Jane (Bond;
Student Identication Number: 0 O S
Course Abbreviation and Number: STAT333
Course Title: Applied Probability
Section(s): 1
Sections Combined Co
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Homework 2
Due: 5 July 2013, in class
1
1. Let X1 , X2 , and X3 be i.i.d. and Bernoulli( 2 ). Consider the following events
E1 = cfw_X1 + X2 = 2
E2 = cfw
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 3
24 May 2013
1. A fair die is rolled repeatedly and independently to get a sequence of numbers.
a) Find the probability that 1 appears before 6
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 4
7 June 2013
1. An insurance company supposes that for each policyholder the number of accidents has a
Poisson distribution, where the mean of
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 5
21 June 2013
1. Find the probability generating function of X NegBin(r, p)
2. What is the sequence generated by A(s)?
es
A(s) =
1s
3. Consider
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 5
14 June 2013
1. Show that
k
n
= (1)n
k+n1
,
n
for any positive integer k .
2. What are the sequences generated by A(s) and B (s), where
A(s) =
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 8
5 July 2013
1. (From Chapter 5 practice problems, with minor modications) Bob and Alice are separately
playing independent (but identical) gam
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 7
28 June 2013
1. Suppose is a renewal event and V = total number of occurrences of in the sequence.
a) If E[V ] = 7, nd the probability that oc
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Tutorial 1
10 May 2013
1. Use the axioms of probability to prove : if two events E and F are independent, so are E
and F .
2. Suppose events E1 and E2 ar
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Quiz 1. Q1
Q2
Q3
Total:
30 May 2013
Name:
Student ID:
1.[8 points] Suppose X1 and X2 are two independent Bernoulli random variables, where
1
P(X1 = 0) =
STAT 333 Assignment 4
Solutions
1. The following problem arises in molecular biology. The surface of a bacterium is
supposed to consist of several sites at which foreign molecules some acceptable
and some not become attached. We consider a particular site
STAT 333 Assignment 4
Due Date: Monday, April 7, 2008
1. The following problem arises in molecular biology. The surface of a bacterium is
supposed to consist of several sites at which foreign molecules some acceptable
and some not become attached. We cons
STAT 333 Assignment 3
Solutions
1. Self-organizing Library Systems
Consider a bookshelf that contains two books, B1 and B2 . The books can be arranged in two possible ways, namely B1 B2 and B2 B1 . Assume that, from time to
time, at epochs n = 0, 1, 2, .
STAT 333 Assignment 3
Due Date: Monday, March 24, 2008
1. Self-organizing Library Systems
Consider a bookshelf that contains two books, B1 and B2 . The books can be arranged in two possible ways, namely B1 B2 and B2 B1 . Assume that, from time to
time, at
STAT 333 Solutions to Assignment 2
1. Let X be a random variable with pgf P (z ). Find the pgfs of X + 1, X + r, 2X and
kX , where r, k are positive integers.
S1
E(z X +1 )
E(z X +r )
E(z 2X )
E(z kX )
z E(z X ) = zP (z )
z r E(z X ) = z r P (z )
Ecfw_(z
STAT 333 Assignment 2
Due Date: Friday, February 15, 2008
1. Let X be a random variable with pgf P (z ). Find the pgfs of X + 1, X + r, 2X and
kX , where r, k are positive integers.
2. (a) Let the random variable X have a geometric distribution with pf f
STAT 333 Assignment 1
1. Two fair dice are thrown. Let X be the score on the rst die, and Y be the larger
of the two scores.
(a) Write down a table showing the joint distribution of X and Y .
(b) Find E(X ), E(Y ), Var(X ), Var(Y ), and Cov(X, Y ).
S 1. (
STAT 333 Assignment 1
Due Date: Wednesday, January 30, 2008
1. Two fair dice are thrown. Let X be the score on the rst die, and Y be the larger
of the two scores.
(a) Write down a table showing the joint distribution of X and Y .
(b) Find E(X ), E(Y ), Va
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Homework 1
Due: 7 June 2013, in class
1. For two RVs X and Y ,
a) Prove that if X and Y are independent, then
E[XY ] = E[X ] E[Y ]
(1)
b) Give an example
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Homework 3
Due: 23 July 2013, in class
1. Two players each toss an unbiased coin independently n times. Show that the probability
that each will have the
Solutions to Quiz 1, STAT 333
1
1. Note that X1 and X2 are independent Bern( 2 ). The indicator random variable
IX1 =X2 also takes its values in cfw_0, 1. By direct enumeration,
1
P(IX1 =X2 = 0) = P(IX1 =X2 = 1) =
2
We rst check the independence of IX1 =
University of Waterloo
Department of Statistics and Actuarial Science
STAT 333 - Spring 2013
Handout: Midterm Exam
25 June 2013, from 2:30 to 3:50
Student Name (PRINT):
Waterloo Student ID Number:
Remark 1: You are allowed to use ve pages of personal note
STAT 333
Midterm Solutions February 2008
AIDS: Calculator, Formula sheet
DURATION: 60 minutes
1. Each part is worth 2 marks
(a) Let I1 , . . . , In be independent indicator random variables (rvs) such that E(Ij ) =
p, j = 1, . . . , n. Find the probabilit