5.1 Random Variables
CHAPTER 5
Discrete Random Variables
This chapter is one of two chapters dealing with random variables. After introducing the notion of a random variable, we discuss discrete random variables:
continuous random variables are left to th
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
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
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
[email protected] (This is the best way to contact D
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: [email protected]
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,