Economic Statistics - Set Theory and Probability for Beginers

7/31/2019 Economic Statistics - Set Theory and Probability for Beginers

1/35


2/35


3/35

UNIVERSITY OF ILLINOISDepartment of Economics

Course: Econ 506, Fall 2012 August 28, 2012Instructor: Anil K. Bera ([email protected]), 225E DKHClass Hours: 1:30 - 3:10 TuTh

Class Room: 215 DKHOffice Hours: 12:00 - 1:00 TuThTA: Yu-Hsien Kao ([email protected])

Prologue

April 1242Baghdad, Iraq

Baghdad took no note of the arrival of Shams (Sun) of Tabriz, a wondering SufiSaint, from Samarkand to the citys famous Dervish Lodge. Shams told the master ofthe lodge, Baba Zaman, that he wanted to share his accumulated knowledge to the

most competent student. Why? Because, Shams said, Knowledge is like brackishwater at the bottom of an old vase unless it flows somewhere. Baba Zaman got seriousand asked a bizarre question: You say you are ready to deliver all your knowledgeto another person. You want to hold the Truth in your palm as if it were a preciouspearl and offer it to someone special. That is no small task for a human being. Arenot you asking too much! What are you willing to pay in return?

Raising an eyebrow, Shams of Tabriz said firmly, I am willing to give my head.

This is an introductory course in mathematical statistics, and its purpose is toprepare you for the econometrics course, Econ 507 (Spring 2013). To carry out agood applied econometrics study, it is necessary to master the econometric theory.Econometric theory requires a good knowledge of statistical theory which in turn has

its foundation on probability theory. Finally, one cannot study probability withoutset theory. Therefore, we will begin at the beginning. We will start with the settheory, and discuss probability and the basic structure for statistics. Then we willslowly move into different probability distributions, asymptotic theory, estimationand hypothesis testing.

After doing all these, the whole course will be just like a candle. It will provideus much valuable light. But let us not forget that a candle will help us to go from oneplace to another in the dark. If we, however, forget where we are headed and insteadconcentrate on the candle, what good will it be?

As you have guessed the course materials will be highly theoretical. No statisticalbackground will be assumed. However, I will take it for granted that you alreadyknow differential and integral calculus and linear algebra. Good Luck!

Course Outline:

1. Introduction

(a) Why statistics?

(b) Statistical data analysis: Life by numbers

2. Probability Theory

(a) Algebra of sets

(b) Random variable

1


4/35

(c) Distribution function of a random variable

(d) Probability mass and density functions

(e) Conditional probability distribution

(f) Bayes theorem and its applications

(g) More on conditional probability distribution(h) Mathematical expectation

(i) Bivariate moments

(j) Generating functions

(k) Distribution of a function of a random variable

3. Univariate Discrete and Continuous Distributions

(a) The basic distributionhypergeometric

(b) Binomial distribution (as a limit of hypergeometric)

(c) Poisson distribution (as a limit of binomial)

(d) Normal distribution(e) Properties of normal distribution

(f) Distributions derived from normal (2, t and F)

(g) Distributions of sample mean and variance

4. Asymptotic Theory

(a) Law of large numbers

(b) Central limit theorems

5. Estimation

(a) Properties of an estimator(b) Cramer-Rao inequality

(c) Sufficiency and minimal sufficiency

(d) Minimum variance unbiased estimator and Rao-Blackwell theorem

(e) Maximum likelihood estimation

(f) Nonparametric method and density estimation

6. Hypothesis Testing

(a) Notion of statistical hypothesis testing

(b) Type I and II errors

(c) Uniformly most powerful test and Neyman-Pearson lemma

(d) Likelihood ratio (LR) test

(e) Examples on hypothesis testing

(f) Raos score or the Lagrange multiplier (LM) test

(g) Wald (W) test

2


5/35

Recommended Text:

A First Course in Probability and Statisticsby B.L.S. Prakasa Rao, 2008, WorldScientific.

However, I will not follow this book closely. For your convenience detailed notes(in four volumes) on the whole course will be made available in the course web-

page. As you will notice, the lecture notes, given the subject matter, are verydry and mechanical. We will try to make things more lively by analyzing someinteresting data (some even depicting your lives) sets and contemporary realworld problems.

Yu-Hsien Kao, TA for this course will meet with the class on Fridays, 1:30 -2:50pm, 215 DKH. Her office hours will be: 11:00am - 12:30pm Mondays.

Course Webpage: Please check Compass regularly for Announcements/ Updateson Homeworks, Exams etc.

Assessment: There will be two closed book examinations. You will also receive fourhomework assignments. The grading of the course will be based on:

Homework 20%First Exam (around mid-semester on a Th) 40%Second Exam(on the last day of the class) 40%

EpilogueIn late October of 1244 in Konya, Turkey, Shams found the student he was looking

for: Jalaluddin Rumi, already a famous Islamic scholar in Turkey. Under the tutelageof Shams, Rumi became one of the most revered poets in the world, as Rumi said, I was raw. I was cooked. I was burned.

March 1248Konya, Turkey

Rumis son Aladdin hired a killer who did not require much convincing.It was a windy night, unusually chilly for this time of the year. A few nocturnal

animals hoofed and howled from afar. The killer was waiting. Shams of Tabriz cameout of the house holding an oil lamp in his hand and walked in the direction of thekiller and stopped only a few steps away from the bush where the killer was hiding.

It is a lovely night, isnt it? Shams asked.Did he know the killer was there? Soon six others joined the killer. The seven

of them knocked Shams to the ground, and the killer pulled his dagger out of hisbelt......

Together they lifted Shams body which was strangely light, and dumped him intoa well. Gasping loudly for air, each of them took a step back and waited to hear thesound of Shams body hitting the water.

It never came.

Taken from: Elif Shafak (2010), The Forty Rules of Love, Penguin Books.

3


6/35


7/35


8/35


9/35


10/35


11/35


12/35


13/35


14/35


15/35


16/35


17/35


18/35


19/35


20/35


21/35


22/35


23/35


24/35


25/35


26/35


27/35


28/35


29/35


30/35


31/35


32/35


33/35


34/35


35/35

Economic Statistics - Set Theory and Probability for Beginers

Documents

Transcript of Economic Statistics - Set Theory and Probability for Beginers