University of Waterloo
STAT 333 Midterm
Term: Fall
Year:
2012
Student Name (Print):
UW Student ID Number:
UW Student Userid:
Instructor (Circle one):
1. Pengfei Li
2. Mehdi Molkaraie
Course Abbreviation and Number:
Course Title:
Stat 333
Applied probabili
Question 1. (8pts) Suppose that 8 cards, of which 4 are red cards and 4 are green cards, are put at
random into 8 envelope, of which 5 are red envelope and 3 are green envelope, so that each envelop will
contain a card. Let X be the number of red cards th
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 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 - 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: 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
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: 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 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
STAT 333 Assignment 2 SOLUTIONS
Due: Thursday, Nov. 4 at the beginning of the class
1. let Xi , i = 1, 2, .n, are i.i.d. random variables with uniform distributions on [0, 1], where
n is a positive integer.
(1) Find Pr (X1 X2 ) or Pr (X1 = min(X1 , X2 ).
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
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
., Question 1. (10pts)
A rat moves through the maze shown in the following picture. At each step, the rat makes a move from
one compartment to another, by choosing passageways with equal probability. For example, if there are
two passageways in the compar
Question 1. (10pts) Consider a sequence of Bernoulli trials in which the outcome is S or F on each
trial. Suppose the probability of S on any trail is 0.5. We are interested in the event AzSFSF. Let TA
be the waiting time for the rst A.
(a) (2pts) Use any
o i 2 A1" [U
(a! '0 f I o o 0
f: 2 2 0 V1 0 0
,- 0 IV). 0 0 O
I 0 o :I. ,0 f
06; <$>2<:= ~.>A,;<_;,U
32W Wallof
7/+,;,;cfw_2#5;3%>
(b) E:(7E,7T,r-J Er)
: M14 5! l 5 llo r.) 77,:2776
77% 12 _. P shamed/7 =7 77, 27mm
5) WIN/la: we Fmgljfj:27fo 1%:
Q E : :7
Question 1. (10pts)
(a) (7pts) A mouse is at the center of a maze. Two doors lead out of the maze. Door 1 leads back to
the center after 4 minutes of scampering. Door 2, however, after 2 minutes of scampering, splits into
two tunnels: Tunnel A and Tunnel
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, . .
STAT 333 - Fall 2014
Applied Probability
Instructors:
Pengfei Li
Martin Lysy
M3 4021
M3 4203
[email protected][email protected]
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
Question 1. (10pts) Suppose X1, X2, and Y are three independent random variables. The probability
density functions for both X1 and X2 are given by f (:13) and the probability density function of Y is given
by g(y), where
2:1: O<cc<1 3y2 0<y<1
= d : .
f(a
University of Waterloo
STAT 333 Midterm
Term: Fall
Year:
2012
Student Name (Print):
UW Student ID Number:
UW Student Userid:
Instructor (Circle one):
1. Pengfei Li
2. Mehdi Molkaraie
Course Abbreviation and Number:
Course Title:
Stat 333
Applied probabili
Solutions of Term test 1
Multiple choice problem: Solutions
1.D
2.C
3.B
4.B
5.B
6.C
Multiple choice problem: comments
(1) Please check the course notes for the improper example in Chapter 2. Question 1 and
this example are almost same.
(2) Please check Ch
Outline of STAT 333 Final Exam
Pengfei Li
Department of Statistics and Actuarial Science
University of Waterloo
2016, Spring term
Pengfei Li (University of Waterloo)
Outline of STAT 333 Final Exam
2016, Spring term
1/9
Important information for final exam