STAT 333: Applied Probability
Homework 1
Instructions: Due Friday, February 6 at 5pm in the designated drop box. Include the names of all collaborators.
1. Suppose that there are K types of butteries in a forest. A scientist goes to the forest
and capture
STAT 333: Applied Probability
Homework 2
Instructions:
Due Friday, March 13 at 5pm in Drop Box 15, Slots 2 & 4.
Include the names of all collaborators.
1. Let X1 , . . . , X4 be discrete random variables with Xn cfw_1, 2, . . . , 5, and denote
p(i, j, k,
STAT 333: Applied Probability
Midterm 1: Practice Questions
1. Let X be a random variable with Binomial distribution with parameters n = 3, p = 0.4,
i.e.
3
P (X = k) =
(0.4)k (1 0.4)3k when k = 0, 1, 2, 3.
k
Let Y = (X 2)2 .
a. Find the probability mass f
STAT 333: Applied Probability
Homework 1: Solutions
K
j=1
1. Let Ij = 1 if buttery type j is caught, and Ij = 0 otherwise. Then X =
and
K
K
j=1 Ij
E[X] = E
=
E[Ij ] = KE[I1 ] = KP (I1 = 1),
j=1
where the second-last equality follows by symmetry, i.e., eac
STAT 333: Applied Probability
Homework 3
Instructions:
Due Monday, April 6 at 5pm in Drop Box 15, Slots 2 & 4.
Include the names of all collaborators.
1. An auto insurance company receives claims from clients according to a Poisson sprocess
with rate = 50
University of Waterloo
STAT 333: Midterm II
Term: Winter
Year:
2015
Student Name (Print):
UW Student ID Number:
UW Student Userid:
Course Abbreviation and Number:
Course Title:
Section(s) (please circle one):
Sections Combined Course(s):
Section Numbers o
STAT 333: Applied Probability
Practice QuestionsPoisson Processes and
Continuous-time Markov Chain
1. Three customers A, B and C enter a bank. A and B to deposit money and C to buy a
money order. Suppose that the time it takes to deposit money is exponent
University of Waterloo
STAT 333: Midterm I
Term: Winter
Year:
2015
Student Name (Print):
UW Student ID Number:
UW Student Userid:
Course Abbreviation and Number:
Course Title:
Section(s) (please circle one):
Sections Combined Course(s):
Section Numbers of
STAT 333: Applied Probability
Homework 2
Instructions:
Due Friday, March 13 at 5pm in Drop Box 15, Slots 2 & 4.
Include the names of all collaborators.
1. Let X1 , . . . , X4 be discrete random variables with Xn cfw_1, 2, . . . , 5, and denote
p(i, j, k,
STAT 333: Applied Probability
Homework 3
Instructions:
Due Monday, April 6 at 5pm in Drop Box 15, Slots 2 & 4.
Include the names of all collaborators.
1. An auto insurance company receives claims from clients according to a Poisson sprocess
with rate = 50
STAT 333: Applied Probability
Tutorial 6
1. Let cfw_Xn n=0,1,. be a discrete-time Markov chain with state space S = cfw_0, 1, 2 and
transition matrix
0 1 2
0 0 0 1
1
P = 1 1 4 1 .
2
4
1
2 1 4 1
2
4
a. Find the communicating classes of this Markov chain, a
STAT 333: Applied Probability
Syllabus
Course Information
Instructor:
Oce:
Contact:
Section I
Yi Shen
M3 4018
yi.shen@uwaterloo.ca
Section II
Martin Lysy
M3 4203
mlysy@uwaterloo.ca
Location:
Time:
Tutorials:
Oce Hours:
MC 2017
MTh 2:30pm - 3:50pm
T 11:30a
STAT 333: Applied Probability
Tutorial 7
1. (Gamblers ruin) A gambler plays a game repeatedly. After each round she gains one
1
dollar with probability p (0, 1), p = 2 , and loses one dollar with probability q = 1 p,
independent from other rounds. Let Xn
Stat 333
Ehrenfest Urn Model of Molecular Diusion
The simplest statement describing the process of diusion is to say that molecules tend to move
from regions of high concentration to regions of low concentration. Imagine a drop of black ink
being placed i
AFM 475 - Midterm Exam Formula Sheets - Winter 2014
Future value of an investment of C today after T years:
C (1 + rm /m)mT = CerT
Converting between discretely and continuously compounded interest rates:
1+
rm
m
m
= er r = m ln 1 +
rm
m
and rm = m er/m
AFM 475
Practice Problem Set #1
Winter 2014
1. Discuss in general terms the risks faced by an investor who has purchased a callable bond, as compared
to an otherwise identical bond which is not callable.
2.
(a) A bank quotes an interest rate of 3.6% per y
AFM 475
Suggested Solutions to Practice Problem Set #1
Winter 2014
1. From the investors perspective, there are several risks from a callable bond, compared with an otherwise identical non-callable bond. First, the cash ows of the bond are not known with
AFM 475
1.
Suggested Solutions to Practice Problem Set #2
Winter 2014
(a) P = $10,000 [1 .0412(240/360)] = $9,725.33.
[1 +YB (240/365 1/2)] [1 +YB /2] $9,725.33 = $10,000
YB = 4.27%
Note that YB can be found either numerically, or using the formula for t
Econ 321 Mid-Term
Instructions:
*Problems should be answered as completely as possible. Partial credit on problems is only
possible if I can locate errors in your calculations so show me your work
*Neatness and organization of your answers is essential
Pa
STAT 333: Applied Probability
Tutorial 5
1. Recall the buttery collection problem in Assignment 1: A scientist goes to the forest
and captures butteries. Suppose that each type of buttery is independently captured
with probability 1/K, and let Xn cfw_1, .
STAT 333: Applied Probability
Tutorial 1
1. Let A and B be two events.
i. Prove that if A and B are independent, then A and B c , Ac and B, Ac and B c are
also independent.
ii. Prove that if P (B) = 0, then A and B are independent.
iii. Prove that if P (B
STAT 333: Applied Probability
Practice Questions 2: Solutions
1. Let Xn be the coin number on the nth day. Then the state space is S = cfw_1, 2. Further
P (X0 = 1) = P (X0 = 2) = 0.5. We need to nd P (X3 = 1).
The transition matrix is
0.7 0.3
0.6 0.4
P=
a
STAT 333: Applied Probability
Midterm 2: Practice Questions
1. Suppose that coin 1 has probability 0.7 of coming up heads, and coin 2 has probability
0.6 of coming up heads. If the coin ipped today comes up heads, then we select coin 1 to
ip tomorrow, and
STAT 333: Applied Probability
Practice Questions 1solutions
1. i. Since X can only take value 0, 1, 2, 3, Y = (X 2)2 can only be 0, 1, 4. P (Y =
0) = P (X = 2) = 0.288, P (Y = 1) = P (X = 1 or X = 3) = 0.496, P (Y = 4) = P (X =
0) = 0.216, P (Y = k) = 0 f
STAT 333: Applied Probability
Tutorial 8
1. A gas station has two pumps. Suppose cars enter the station according to a Poisson
process with rate . If a car arrives when a pump is free, it will use the pump to ll up
on gas for an exponential time with rate
STAT 333: Applied Probability
Tutorial 9
1. Let X(t) be a Markov process on S = cfw_0, 1, 2 with generator matrix
0
1
2
0 0.4 0.2
0.2
R = 1 0
3
3 .
2 0.25 0.5 0.75
a. Is this Markov process a Birth-Death process?
b. Suppose that X(3) = 0. What is the prob
STAT 333: Applied Probability
Tutorial 9
1. Let X(t) be a Markov process on S = cfw_0, 1, 2 with generator matrix
0
1
2
0 0.4 0.2
0.2
R = 1 0
3
3 .
2 0.25 0.5 0.75
a. Is this Markov process a Birth-Death process?
Solution: A Birth-Death process has the co
STAT 333: Applied Probability
Markov Chains: The Branching Process
The random walk is an innite state Markov chain with important applications in nance.
Branching processes are another type of innite state Markov chain with important applications in biolo
STAT 333: Applied Probability
Conditional Expectations
version: 2015-01-30 21:04:03
We saw that the conditional distribution of a random variable X can be used to make
probability statements exactly the same way a regular distribution would. The only dier