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ECON 41 - PROBABILITY FOR ECONOMISTS

(Winter, 2024)

Description: This course is an introduction to the probability theory with an emphasis on its use in economics. It introduces basic concepts such as: sets and events; probability, conditional probability, and independence; discrete and continuous random variables; integration and expectation; law of large numbers and central limit theorem, confidence intervals, and statistical hypothesis test.

Textbook: A Brief Course in Mathematical Statistics by Elliot A. Tanis and Robert V. Hogg (Prentice Hall). We do not use the entire book.

Course Outline:

Week 1-2. Course Introduction and Basic Concepts of Probability Theory

•      Course introduction, Materials to be Covered and the Grading Policy

•      Sample Space and the Algebra of Sets

•     The Probability Function

•     Method of Enumeration

Week 2-3. Conditional Probability and Independence

•      Conditional Probability

•     Independence

Week 4-5. Discrete Random Variables

•     Probability Mass Function and Cumulative Distribution Function

•     Mean and Variance of a Discrete Random Variable

•     Expectations

•     Binomial Random Variable

•     Poisson Random Variable

•     Mid-term Exam

Week 6. Multivariate Discrete Random Variables

•     Joint Probability Mass Function

•     Independence

•      Covariance and Correlation

•     Linear Functions of Random Variables

Week 7. Continuous Random Variables

•     Probability Density Function (PDF)

•      Cumulative Distribution Function (CDF)

•     Mean and Variance of a Continuous Random Variable

•     Expectations

Week 8. Normal Distribution

•     PDF and CDF of Normal Distribution

•     Mean and Variance of Normal Random Variables

•     Linear Transformation of Normal Random Variables

•     Quantiles of Normal Distribution

Week 9. Law of Large Numbers and the Central Limit Theorem

•     Chebyshev’s Inequality

•     Law of Large Numbers

•     Central Limit Theorem

Week 9- 10. Confidence Intervals

•     Interval Estimation of Parameters

•     Confidence Interval of Normal Mean

Week 10 (Tentative). Hypothesis Testing

•      Statistical Hypothesis and Decision Rule

•      Type-I and Type-II Errors

•     Testing the Normal Mean

Homework: Homework problems are listed at the end of slides of each topic. They are not required but some problems in the midterm and final exams are revised from the homework problems. So, you should really understand how to solve these problems. Your TA will go over the homework problems in the TA session.

Exams: There will bean in-class mid-term exam on Feb 21, 2024, and a final exam. The date of the final exam is decided by the university, and it is your responsibility to find the final exam schedule and location. (Last time I checked the registrar’s webpage, it is administered on 8:00 am -- 11:00 am, Monday, March 18, 2024).  Here are some important policies about the exams:

1.    The midterm will cover everything discussed in lecture 1 to lecture 7. The final exam will cover everything discussed this quarter, including the material covered in the midterm exam.

2.    In both exams, Tables in the appendix of the textbook, when needed, will be provided.

3.    The exams will be of the multiple-choice format, and you are required to use the Scantron. You cannot assess and revise the Scantron after you turn it in, even if there are unfinished questions.

4.    If you erase an answer on your Scantron, you have to write the letter answer over to the right-hand side of the Scantron on the appropriate line.

5.    In both exams, you are required to bring a calculator. The only acceptable calculator is Canon LS- 100TS. It can be bought from the ASUCLA bookstore.

6.    No cheat sheet is allowed in both exams.

7.    Cellphones, PDAs or any other electronics are not allowed during the exams.

8.    In both exams, you should write down your name and UID in both the Scantron and the exam paper. The exam paper should be turned in together with the Scantron for the full consideration of your grade.

Evaluation: Your final letter grade in this course will be determined by the weighted average of your midterm exam and final exam scores. No other factors will be considered in the grading process. There are two weighting schemes. In the first scheme, the midterm exam and final exam are weighted at 30% and 70%, respectively. In the second scheme, the final exam is given 100% weight. The grading will follow a curve, with approximately 25% of students receiving A's, 35% receiving B's, and 25% receiving C's. To address ties and ensure precision, a mathematical algorithm will be employed as described below:

Calculate weighted averages: Your midterm score (M) is divided by the maximum possible score and multiplied by 100. Your final score (U) is divided by the maximum possible score and multiplied by  100. Two weighted averages are obtained: W1=M×0.3+U×0.7 and W2=U.

Rank determination (for both W1 and W2): Define r1 as the number of students in the class with W1 strictly higher than your W1, and R1 = r1 + 1. Similarly, definer2 as the number of students in the class with W2 strictly higher than your W2, and R2= r2 + 1.

Preliminary grade assignment: Apply the algorithm below to R1 and R2 separately (R = R1, R2):

1.    Let n1, n2, n3, n4, n5, n6, n7, n8, n9, n10 and n11 denote the smallest integers larger than or equal to 0.05×n,  0.15×n,  0.25×n,  0.35×n,  0.45×n,  0.60×n,  0.70×n,  0.80×n,  0.85×n,  0.90×n  and  0.95×n, respectively. Here, then denotes the totally number of students in the class.

2.    IfR≤n1, you will get A+. If n1

3.    If n3

4.    If n6

5.    If n8

6.    Note: In this context, the symbols < and ≤ have distinct mathematical meanings.

Final Grade Determination: If both your W1 and W2 are less than or equal to 40, your final course grade is F regardless of your preliminary grades. If at least one of your W1 and W2 is strictly larger than 40, then your final course grade is the better one of the two preliminary grades obtained from R1 and R2 above.

Disabled Students and the Center for Accessible Education (CAE): Any student with a pre-existing illness or condition who requests special arrangements must: (a) qualify under CAE rules for such special arrangements and (b) must take the exam with CAE. Any such arrangements with CAE must be made the first week of classes. The instructor must be informed of any such arrangement in the first week of classes. For additional information and the qualification conditions of the Center for Accessible Education, please visit their website at http://www.cae.ucla.edu/. All other students must take the exam at the scheduled time under the same time constraints. It is the responsibility of all students who request special arrangements with CAE to be familiar with all of their rules as well as the rules of this class.

Academic Dishonesty: Any cases of academic dishonesty will be reported to the Dean’s office of Students. For more details, please refer to the Dean’s Office of Studentswebsite.

Other Rules: Consult the Common Syllabus, which can be found at:

https://economics.ucla.edu/undergraduate/common-syllabus/

Rules governing make-up exams, e.g., can be found there.