Chapter 2
Combinatorial Probability
2.1 Permutations and combinations
As usual we begin with a question: Example 2.1. The New York State Lottery picks 6 numbers out of 54, or more precisely, a machine picks 6 numbered ping pong balls out of a set of 54. H
Department of Economics
University of Toronto
Fall 2012Winter 2013
Course
ECO227Y Quantitative Methods in Economics
Lecturer
Victor Yu
Office
GE344, Max Gluskin House, 150 St. George Street
E-mail
victor.yu@utoronto.ca (This is the best way to contact D
ECO227Y Quantitative Methods in Economics
Department of Economics
University of Toronto
Fall 2011
Dr. Yu
Test 1
Date:
Wednesday October 19, 2011
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages).
Notes:
This test c
ECO227Y Quantitative Methods in Economics
Department of Economics
University of Toronto
Summer 2010
Dr. Yu
Test 1
Date:
Wednesday June 2, 2010
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages).
Notes:
This test con
ECO227Y Quantitative Methods in Economics
Department of Economics
University of Toronto
Summer 2009
Dr. Yu
Test 1
Date:
Monday June 1, 2009
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages).
Notes:
This test consis
ECO227Y Quantitative Methods in Economics
Department of Economics
University of Toronto
Summer 2008
Dr. Yu
Test 1
Date:
Wednesday June 4, 2008
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages).
Notes:
This test con
Getting CDF from PDF
Thomas Laetsch
Given a probability density function (pdf, or just density function), p( x), we have the following
properties:
1.
p( x)dx = 1
2. p( x) 0 always
Now, given a cumulative distribution function (cdf), P( x), we have the pro
ECO 227Y1
Proposed Solutions to Term Test 4
12 April 2011
1. The natural logarithms of the hourly wages in a particular industry are
known to be normally distributed with mean 2.5802 and standard deviation
.9163. A company in this industry employs 40 work
r st ts r Prr
tr
st
stts s ss sttsts tst r tts tst t
t rt tt ts stt sss t sttsts tst s t tts tst
s r t sr t rst rs
t rt ts stt s t tsts rs
ts stt s t tts tst
r st ts r Prr
r
st
rs X Y t t
fX,Y (x, y) = 2e2xy ,
x > 0 y > 0
tr tr r t X Y r t rs
t W = X + 2Y t r W rs
t
rst t r s X Y
fX (x) =
fX,Y (x, y)dy =
2e2xy dy
ey dy
= 2e2x
0
0
0
= 2e2x (ey )|
0
= 2e2x ( lim ey e0 )
y
= 2e
2x
= 2e
2x
fY (y) =
fX,Y (x
ECO 227Y1
Proposed Solutions to Term Test 3
8 March 2011
1. Let X1 , . . . , Xn be iid N (, 2 ) random variables.
(a) Suppose c1 , . . . , cn are known constants. Let Z =
the density of Z.
The mgf of Z is
)]
[
( n
ci X i
mZ (t) = E exp t
n
i=1 ci Xi .
i=1
ECO 227Y1
Sample Examination Questions
12 October 2010
1. Two six-sided dice are thrown sequentially, and the face values that come
up are recorded.
(a) List the sample space.
cfw_(n1 , n2 ) : n1 , n2 cfw_1, . . . , 6.
(b) List the elements that make up
Do the following Exercises
2.2.4, 2.3.4, 2.4.6, 2.4.28, 2.4.52, 2.5.2, 2.5.52, 2.6.26, 2.6.34, 2.6.52, 2.7.8, 3.2.4,
3.3.10, 3.4.14
Solutions for Problem Set 1
ECO 227Y1
Proposed Solutions to Term Test 2
20 December 2010
1. Aircraft arrive or depart Toronto Pearson International Airport according
to a Poisson process with a rate of 42 per hour. An excessive number of
aircraft arriving or departing during a brief
ECO 227Y1
Sample Examination Questions
7 December 2010
1. Below are the last three lines of Ozymandias, a famous short poem by Percy
Bysshe Shelley (17921822):
Nothing beside remains. Round the decay
Of that colossal wreck, boundless and bare
The lone and
ECO 227Y1
Proposed Solutions to Term Test 1
19 October 2010
1. The one-year returns, in percent, for a sample of 100 midcap mutual funds
has a distribution with a mean of -.02 and a standard deviation of .87. Approximately how many of the funds would you
Information for Test 1
ECO227Y Quantitative Methods in Economics
University of Toronto
Fall 2012
Time
2:00pm 4:00pm, Wednesday October 17, 2012
Location
EX310, Examination Facility, 255 McCaul Street
Aids Allowed
Any calculator and one aid sheet (8.5x11)
Introduction
Chapter 2
Chapter 3
ECO227 Tutorial 3
Eric H. Mak
Department of Economics
University of Toronto
Last Updated: October 9, 2012
Eric H. Mak
ECO227 Tutorial 3
U of T
Introduction
Chapter 2
Chapter 3
Email: honam.mak@utoronto.ca
Homepage: https:/
Introduction
Law of Total Probability and Bayes Rule
Independence
Combinatorics
Combinatorial Probability
ECO227 Tutorial 2
Eric H. Mak
Department of Economics
University of Toronto
Last Update: October 3, 2012
Eric H. Mak
ECO227 Tutorial 2
U of T
Introdu
Introduction
Sample Space and the Algebra of Sets
The Probability Function
Conditional Probability
ECO227 Tutorial 1
Eric H. Mak
Department of Economics
University of Toronto
20th September, 2012
Eric H. Mak
ECO227 Tutorial 1
U of T
Introduction
Sample Sp
ECO227Y Quantitative Methods in Economics
Summer 2010
Department of Economics
University of Toronto
Dr. Yu
Test 2
Date:
Wednesday July 7, 2010
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages) prepared by
student.
N
ECO227Y Quantitative Methods in Economics
Summer 2009
Department of Economics
University of Toronto
Dr. Yu
Test 2
Date:
Monday June 29, 2009
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages) prepared by
student.
Not
ECO227Y Quantitative Methods in Economics
Summer 2008
Department of Economics
University of Toronto
Dr. Yu
Test 2
Date:
Wednesday, July 2, 2008
Time allowed:
Two (2) hours
Aids allowed:
Calculator and one aid sheet (two 8.5x11 pages) prepared by
student.
Sample Test 1
ECO227 Quantitative Methods
Question 1. A certain city has one morning newspaper and one evening newspaper. It is
estimated that 30% of the citys household subscribe to the morning newspaper and 45%
subscribe to neither newspaper. Besides, 4
Continuous Random Variables
October 8, 2010
Continuous Random Variables
Continuous Random Variables
Many practical random variables are continuous. For example:
1
The speed of a car;
2
The concentration of a chemical in a water sample;
3
Tensile strengths
Discrete Random Variables
October 7, 2010
Discrete Random Variables
Random Variables
In many situations, we are interested in numbers associated with
the outcomes of a random experiment. For example:
Testing cars from a production line, we are interested
Probability Tables
STA 281 Fall 2004
1
Introduction
We have previously talked about probability using mathematical notation such as P (A), P (A B), P (A B c ) and so on. We derived several theorems to describe how these probabilities relate to each other,