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
Stat 333 Homework 1
Instructions: Due Thursday, Oct 8 at 6:30pm in drop box # 14. Questions
1-6 are worth 5 points each.
1. Suppose that there are K types of butteries in a forest. A scientist goes to the forest
and captures n butteries. Suppose that each
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 #3
Due: Dec. 1st in Pengfeis oce no later than 4pm.
Note: Please use the cover page for the assignment. No cover page=0%.
1. Consider the Markov chain with the state space S = cfw_0, 1, 2, 3, 4 and the transition matrix
0
0
1
P = 2
3
STAT 333: Applied Probability
Homework 1
Instructions:
Due Tuesday, October 18 at 5pm in Drop Box 15, Slots 1-3 for Section 1, Slots 4-6 for Section 2.
Include the names of all collaborators.
1. Prove the following properties of probability:
a. Let E be a
STAT 333 - Spring 2014 - Assignment 1
Due Thursday June 5 at start of class
This assignment may be completed in pairs. Only one person should submit the paper. Both
names and ID numbers should be on it. Both will receive the same grade.
1. Five people are
Stat 333 Homework 1
Solution
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., each of the buttery types
Stat 333 Homework 3
Instructions: Due Monday, Nov 30 at 6:30pm in drop box # 14. Questions
1-6 are worth 5 points each.
Print and attach this page as your cover page. Failure to do so will result in
a credit of zero.
Last Name:
First Name:
Section
Student
Chapter 2
Poisson Processes
2.1 Exponential Distribution
To prepare for our discussion of the Poisson process, we need to recall the
denition and some of the basic properties of the exponential distribution. A random
variable T is said to have an exponent
1
Essentials of Stochastic Processes
Rick Durrett
Version Beta of the 2nd Edition
August 21, 2010
Copyright 2010, All rights reserved.
2
Contents
1 Markov Chains
1.1 Denitions and Examples . . . .
1.2 Multistep Transition Probabilities
1.3 Classication of
STAT 333 Assignment 1 SOLUTIONS
1. Consider 10 beads on a bracelet having probability p of being blue and probability 1 - p of
being purple. We say a changeover occurs whenever a bead is a different colour from the one
beside it. For example, if the beads
STAT 333 - Fall 2014
Applied Probability
Instructors:
Pengfei Li
Martin Lysy
M3 4021
M3 4203
pengfei.li@uwaterloo.ca
mlysy@uwaterloo.ca
Oce hours: Will be posted on the course website or announced by the instructors.
Course Webpage: learn.uwaterloo.ca (Lo
STAT333 Fall 2014 Tutorial #4
1. A manuscript is sent to a typing rm consisting of typists A, B, and C. If it is typed by A,
the number of errors made is a Poisson random variable with mean 2; if it is typed by B, the
number of errors made is a Poisson ra
STAT 333: Assignment 2
Due: Nov. 7th in class, or in Pengfei or Martins oce no later than 4pm on Nov. 8th.
Note: Please use the cover page for the assignment. No cover page=0%.
1. Let X1 , X2 , X3 , . . . be independently and identically distributed Berno
STAT 333: Assignment 1
Due: October 9th in class, or in Pengfei or Martins oce no later than 5pm on October 10th.
Note: Please use the cover page for the assignment.
1. A collection of n coins is ipped. The outcomes are independent and the ith coin comes
STAT 333 Assignment 3 SOLUTIONS
1. At all times, a container holds a mixture of N balls, some white and the rest black. At each
step, a coin having probability p, 0 < p < 1, of landing heads is tossed. If it is heads, a ball is
chosen at random from the c
Solutions for Practice Questions in Chapter 2.
1. Let In = 1 if an record occurs at step n, otherwise 0, n = 2, 3, 4, . . . .
(a) Note that P (a record at time n) = P (Xn = maxcfw_X1 , . . . , Xn ). Since X1 , . . . , Xn are
iid continuous r.v.s, then eac
STAT 333 Assignment 3
Due: Monday April 2 at the beginning of class
1. At all times, a container holds a mixture of N balls, some white and the rest black. At each
step, a coin having probability p, 0 < p < 1, of landing heads is tossed. If it is heads, a
Birth and death process examples
1. (Single server problem) Students arrive at T.H. at a rate (Poisson rate or exponential rate) of per minute. Assume only one person serves in T. H. Suppose the
service time for each student is exponential distributed wit
STAT333 Fall 2012 Tutorial #2
1. Consider a sequence of Bernoulli trials in which the outcome is S or F on each trial.
Suppose the probability of S on any trial is p with 0 < p < 1. Let T = be the waiting
time for 1st SF.
(a) Find P (T = k), k = 2, 3, . .
STAT333 Fall 2014 Tutorial #3
1. (Example 3.2 in supplementary notes) Suppose we have t = n + m Bernoulli trials and
the probability of success in each trial is P (S) = p with 0 < p < 1. Let
X = number of success in t Bernoulli trials,
Y = number of succe
24. A coin, having probability p of coming up heads is flipped until at least one
head and one tail have been flipped.
a) Find the expected number of flips needed
b) Find the expected number of flips that land on heads
c) Find the expected number of flips
STAT 333 Assignment 2
Due: Friday, March 2 at the beginning of class (or up to 10 minutes in)
1. Consider a sequence of independent tosses of a fair coin. Each toss results in a head H or a tail T.
Let be the event that the total number of H equals exactl
STAT 333 Assignment 2 SOLUTIONS
1. Consider a sequence of independent tosses of a fair coin. Each toss results in a head H or a tail T.
Let be the event that the total number of H equals exactly one-third the number of tosses, i.e.
we say occurs on the nt
Stat 333: Test 2 - Winter 2012 SOLUTIONS
1. Suppose an innite sequence of letters is selected randomly from the 26-letter alphabet.
(a) [2] Find the expected # of trials until the rst occurrence of the pattern
EFGEEFGE
E [TEF GEEF GE ] = E [TE ] + E [TEF
Stat 333: Test 1 - Winter 2012 SOLUTIONS
1
1. Suppose we toss a fair coin (P (H ) = 2 ) until we observe one H , and let Y be the
number of tosses required. Then we toss a second biased coin (with P (H ) = p) until
we observe Y heads, and let X be the num
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Student Name (Print):
Signature:
Instructions:
Not all questions and parts are equally difficult.
The back of the pages can be used as scratch papers, but they will not be graded.
You may tear off the formula sheet on the last page.
Marking Scheme:
Que
Student Name (Print):
Signature:
Instructions:
Not all questions and parts are equally dicult.
The back of the pages can be used as scratch papers, but they will not be graded.
You may tear o the formula sheet on the last page.
Marking Scheme:
Question
STAT 333: Applied Probability
Tutorial 9
1. Consider a single server queueing system with abandonments. More precisely, customers arrive according to a Poisson process with intensity > 0. The server has an
exponential service rate > 0. In addition, each c
STAT 333: Applied Probability
Homework 2
Instructions:
Due Friday, November 11 at 5pm in Drop Box 15, Slots 1-3 for Section 1, Slots 4-6 for Section 2.
Include the names of all collaborators.
1.
a. Consider a Markov chain with transition matrix
0
1
2
3
4
STAT 333: Applied Probability
Tutorial 8: Solutions
1. (a) A birth and death process has the condition that only Ri,i+1 and Ri,i1 can be
positive, and so Ri,j = 0 when |i j| 2. However according to the above generator
matrix, R0,2 , R2,0 > 0, so this is n