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

= Answers for Tutorial 5 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. Sampling distribution with sample size n is the probability distribution of all the possible
values of a sample statistics.
2. The CLT simply

Inferences Based on Two Samples
= Lecture 10 =
Inferences Based on Two Samples
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 41
Inferences Based on Two Samples
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Comparing two population means:

= Tutorial 5 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. What is sampling distribution of a sample statistic? Explain with your own words and sketch
the idea on a paper.
2. What is the Central Limit Theorem? Exp

A Brief Introduction on Regression
= Lecture 12 =
A Brief Introduction on Regression
Dr. Liu Zhengning
Email: ecslz@nus.edu.sg
1 / 42
A Brief Introduction on Regression
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Estimator vs. estimate
2
From sampling distributi

= Suggested Answers for Midterm Exam =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1. Define Xi as the time to complete the race by runner i, where i = 1, 2, 3, and 4. We know
2 = 2 = 100.
that Xi = = 60 and X
i
(a) E(X) = E(X1 + X2 + X3 + X

= Answers for Tutorial 9 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. DIY.
2. (a) False. To show that 1 and 0 are good estimators you need use the properties of the
sampling distribution of 1 and 0 . A large R2 o

= Tutorial 8 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. Give a real-world situation in which you will use the paired dierence experiment method to
test a hypothesis.
2. Label each of the following statement tru

= Tutorial 3 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. For a continuous random variable, what are the probability density function and the culmulative distribution function, respectively?
2. In what conditions

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

= Tutorial 2 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. For a discrete random variable, what are the probability density function and the culmulative
distribution function, respectively?
2. In what conditions d

Inferences Based on a Single Sample: Test of Hypothesis
= Lecture 9 =
Inferences Based on a Single Sample:
Hypothesis Testing
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 48
Inferences Based on a Single Sample: Test of Hypothes

= Tutorial 7 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. Give an example of hypothesis testing in your daily life.
2. Consider the hypotheses H0 : = 0 and Ha : > 0
(a) If H0 is rejected in favor of Ha , will it

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

= Answers for Tutorial 6 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. A point estimator of a population parameter is a rule that tells us how to use the sample
data to calculate a single number that can be used t

NATIONAL UNIVERSITY OF SINGAPORE
EC2303: Foundations for Econometrics
Semester 2, AY2013/2014
Time Allowed: 2 Hours
INSTRUCTIONS TO STUDENTS
1. This assessment paper contains NINE (9) questions and comprises
ELEVEN (11) printed pages. There are six (6) ap

NATIONAL UNIVERSITY OF SINGAPORE
EC2303: Foundations for Econometrics
Semester 1, AY2014/2015
Time Allowed: 2 Hours
INSTRUCTIONS TO STUDENTS
1. This assessment paper contains ELEVEN (11) questions and comprises
TWELVE (12) printed pages. There are six (6)

Sampling Distributions
= Lecture 7 =
Sampling Distributions
Dr. Liu Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 38
Sampling Distributions
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
The concept of sampling distribution
2
Mininum-variance unb

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

= Tutorial 9 =
(Self-practice tutorial)
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. Give a real-world situation in which you will conduct a regression analysis.
2. Label each of the following statement true or fal

= Tutorial 6 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. What are point estimator and interval estimator, respectively? Explain with your own words
and sketch the idea on a paper.
1.645
2. How do you interpret

= Answers for Tutorial 1 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer 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

Type I & II Error
Hypothesis tests, variance
Two-sample tests
Chapter 9: Hypothesis Tests of a Single
Population
9.5-9.6, 10.1
Week 11
26 October 2015
Week 11
L11: Hypothesis Tests
1 / 39
Type I & II Error
Hypothesis tests, variance
Two-sample tests
Lectu

Sample size, P
Hypothesis Concepts
Hypothesis Tests, Mean
Hypothesis Tests, Proportion
Chapter 9: Hypothesis Tests of a Single
Population
9.1-9.3
Week 10
19 October 2015
Week 10
L10: Hypothesis Tests
1 / 62
Sample size, P
Hypothesis Concepts
Hypothesis Te

Dependent samples
Ind., x & y known
Ind., x & y unknown
Two population proportions
Chapter 10: Hypothesis Tests of Two
Populations
10.2, 10.3, 10.5
Week 12
02 November 2015
Week 12
L11: Hypothesis Tests
1 / 40
Dependent samples
Ind., x & y known
Ind., x &

Inferences Based on Two Samples
= Lecture 10 =
Inferences Based on Two Samples
Dr. LIU Zhengning
Department of Economics, NUS
AY2016/17, Semester 1
1 / 41
Inferences Based on Two Samples
Liu (NUS): ecslz@nus.edu.sg
Agenda
1
Comparing two population means:

= Answers for Tutorial 7 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. DIY.
2. (a) No. For two-tailed test, /2 will be assigned to each tail. The RR given by |z| > |z/2 |
will be smaller than the RR given by |z| >

= Tutorial 1 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer 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

= Tutorial 4 =
(Self-practice 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
cxy dx dy

= Answers for Tutorial 8 =
EC2303 Foundations for Econometrics
AY2016/17, Semester 1
1
Short-answer Questions
1. DIY.
2. (a) True. This is the idea of sample size determination.
(b) False, because we do not know about x
before we do the sampling.
2
Probl