Introduction to Statistics
= Lecture 1 =
Introduction to Statistics
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 50
Introduction to Statistics
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Areas of statistics
2
Fundamental elements of s
= Tutorial 1 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Shortanswer Questions
1. What is selection bias? Could you give an example of such bias you have recently encountered?
2. What is unethical statistical practice? Could you give an
= Answers for Tutorial 4 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1. DIY
2. Since = E(X) =
P
V ar(X) =
x xpX (x),
using the definition of variance,
X
X
(x )2 pX (x) =
(x2 + 2 2x)pX (x)
x
x
=
X
x2 pX (x) +
x
X
2 pX (x)
x
= E(X 2 ) + 2 2
EC2303 Lecture 6
Mean

Discrete RV

o
Continuous RV
o
Finding expected value for a function

Discrete RV
o
Finding E(aX + b) and E(aX + bY)

Constant can be pulled out
o
o
Variance

Discrete RV

o
Continuous RV

o
Formula
o
o
B is 0 as b is a const
Dealing with Two Random Variables
= Lecture 5 =
Dealing with Two Random Variables
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 35
Dealing with Two Random Variables
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Joint probability distribu
Lecture 1. Welcome to EC2303
Juwon Seo
Department of Economics
National University of Singapore
Jan 12, 2016
Juwon Seo (NUS)
Lecture 1. Welcome to EC2303
Jan 12, 2016
1 / 23
Instructor: Dr. SEO Juwon
Lecture: Tues 12:001:35 pm, LT8
Office: AS2 #05 28
Off
Review for the Final Examination
Instructor: Juwon Seo
April 11, 2016
Several tips for the final exam.
There are five questions. Total mark is 50. The exam is a closed book
assessment.
The standard normal distribution table is attached. You may or may n
Lecture 12. Central Limit Theorem
Juwon Seo
Department of Economics
National University of Singapore
April 12, 2016
Juwon Seo (NUS)
Lecture 12. Central Limit Theorem
April 12, 2016
1 / 20
Course outline: after the midterm
Where are we?
Probability foundat
Lecture 9. Conditional Distributions and
Independence
Juwon Seo
Department of Economics
National University of Singapore
Mar 22, 2016
Juwon Seo (NUS)
Lecture 9. Conditional Distributions and Independence
Mar 22, 2016
1 / 21
Course outline: after the midte
Lecture 11. Estimation
Juwon Seo
Department of Economics
National University of Singapore
April 5, 2016
Juwon Seo (NUS)
Lecture 11. Estimation
April 5, 2016
1 / 18
Course outline: after the midterm
Where are we?
Probability foundations:
I Joint distributi
Review for the Final Examination
Instructor: Juwon Seo
April 11, 2016
Several tips for the final exam.
There are five questions. Total mark is 50. The exam is a closed book
assessment.
The standard normal distribution table is attached. You may or may n
L10. Sampling distribution and Sample mean
Juwon Seo
Department of Economics
National University of Singapore
Mar 29, 2016
Juwon Seo (NUS)
L10. Sampling distribution and Sample mean
Mar 29, 2016
1 / 36
Course outline: after the midterm
Where are we?
Proba
Lecture 2. Welcome again to EC2303
Juwon Seo
Department of Economics
National University of Singapore
Jan 19, 2016
Juwon Seo (NUS)
Lecture 2. Welcome again to EC2303
Jan 19, 2016
1 / 31
Change in Office Hours
Office: AS2 #05 28
Office Hours:
Tues 1:353:3
Integration
Ra
a
Rb
f (x)dx = 0
a
f (x)dx =
Rb
Rb
a [f (x) + g(x)]dx =
Rb
a
f (x)dx +
a
Rb
f (x)dx
Rb
a
Rb
g(x)dx
a
f (x)dx =
a
Rc
a
f (x)dx +
Rb
c
f (x)dx
b
f (x)dx = F (x) = F (b) F (a)
a
Descrptive Statistics & Probability
=
PN
CnN =
xi
Pn
N
x
=
N!
= Answers for Assignment 2 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1. n = 340, p = 61/340 = 0.179. The hypotheses are H0 : p 0.25 and Ha :< 0.25. = 0.05.
The test statistic is z = 3.006. Since 3.006 < 1.645 = z0.05 , we reject H0 .
is
Discrete Random Variables
= Lecture 3 =
Discrete Random Variables
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 44
Discrete Random Variables
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Random variables (RVs)
2
Discrete RVs
3
Probabilit
= Tutorial 4 =
(Selfpractice tutorial)
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1. Referring to Example 5.2 on page 13 of Lecture 5. Show that the same answer can be obtained
by taking the integration in the following way:
Z Z
Z
cxy dx d
Classical Probability Theory
= Lecture 2 =
Classical Probability Theory
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 44
Classical Probability Theory
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Events, sample spaces, and probability
2
EC2303 Foundation of Econometrics
EC2303 Foundation of Econometrics
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1/6
EC2303 Foundation of Econometrics
Liu (NUS): ecslz@nus.edu.sg
About the Lecturer
Lecturer: Dr. LIU Zhengning
Perso
EC2303 Lecture 4
Continuous Random Variable


Probability distribution of CRV is a smooth curve or function fX(x). The probability that X falls
between two values, a and b, is the area under the curve between a and b, P (axb)
o Integrate between a and b
EC2303 Lecture 3
Random Variables

(RV) assumes numerical values associated with the random outcomes of an experiment
Discrete Random Variables

Countable, can be finite or infinite
integer
Continuous Random Variable

Uncountable, infinite number of va
= Assignment 1 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
Please write your name, matriculation number, and tutorial group number on
the top of your submission. TWO (2) marks will be deducted if any of the above
information is missing.
= Answers for Tutorial 3 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Shortanswer Questions
1. For a continuous random variable, X, the pdf is a function or curve, denoted as fX (x); the
Rt
cdf is given by FX (t) = P (X t) = fX (x)dx.
2.
EC2303 Lecture 5
Joint Probability Distribution

When single RV are not sufficient to characterize
Probability distribution that defines the simultaneous behaviour of 2 or more random
variables
PDF
o
o
Properties
Integral of the region (R) will give 1 f
= Tutorial 3 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Shortanswer Questions
1. For a continuous random variable, what are the probability density function and the culmulative distribution function, respectively?
2. In what conditions
= Suggested Answers for Assignment 1 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1. The random variable X can be illustrated in Table 1. The pdf of X is therefore pX (x) =
P
P (X = x) = (1 p)x1 p. By definition, E(X) = x x pX (x), where x
EC2303 Lecture 2
Definition








Experiment
o A process that leads to a single outcome that cannot be predicted with certainty
Sample point
o Most basic outcome of an experiment (small letter)
Sample space
o Collection of all its sample points
= Tutorial 2 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Shortanswer Questions
1. For a discrete random variable, what are the probability density function and the culmulative
distribution function, respectively?
2. In what conditions d
= Answers for Tutorial 2 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Shortanswer Questions
1. For a discrete random variable, X, the pdf is denoted as PX (x) = P (X = x); the cdf is
denoted as FX (t) = P (X t).
2. Consider counting the
= Assignment 2 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
Please write your name, matriculation number, and tutorial group number on
the top of your submission. TWO (2) marks will be deducted if any of the above
information is missing. F
Topics on the Mean and Variance
= Lecture 6 =
Topics on the Mean and Variance
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 33
Topics on the Mean and Variance
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Understanding the mean and expec
= Answers for Tutorial 1 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Shortanswer Questions
1. Selection bias happens when a subset of experimental units in the population has little or no
chance of being selected for the sample. As a re
Continuous Random Variables
= Lecture 4 =
Continuous Random Variables
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 40
Continuous Random Variables
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
The pdf and cdf for continuous RVs
2
The nor