(PM 522a - Fall 2002)

Instructor:                       Kiros Berhane">

 

UNIVERSITY OF SOUTHERN CALIFORNIA

DEPARTMENT OF PREVENTIVE MEDICINE

SYLLABUS

 

INTRODUCTION TO THE THEORY OF BIOMETRY PRINCIPLES

(PM 522a - Fall 2002)

 

 

Instructor:                       Kiros Berhane, Ph.D.

                                         Biostatistics Division

                                         CHP 220A, (323) 442-1994

                                        e-mail:              kiros@usc.edu

                     Office Hours:  By appointment only

Teaching Assistant:       TBA

                                         e-mail:     TBN

Office hours: Tuesdays 12-1; Wednesdays 11-12; Room: CHP 217

 

            Sessions:                       Wednesdays and Thursdays (9:00-11:00)

            Room:                            CHP 217

 

Course Objectives:

Part I. To introduce fundamental principles of basic probability theory, common parametric distributions, distributions of functions of random variables, sampling and sampling distributions. At the end of this part, students will be able to understand the probabilistic foundations of biostatistical analysis. Students will also be able to understand the basic theory of distributions and acquire thorough knowledge of common parametric families of distributions. In addition, students will be able understand joint, marginal and conditional distributions, to derive distributions of functions of random variables and to understand the theoretical foundations of sampling and sampling distributions

 

Part II. To introduce fundamental principles of statistical inference and the rationale behind commonly used inferential tools in biostatistics/biometry. At the end of the course, students will be able to understand the basic theoretical foundations for methods of point estimation, properties of estimators and tools for interval estimation. Students will also be able to understand the basic theory of hypothesis testing through a detailed discussion on types of hypothesis testing, optimal properties for hypothesis testing techniques. Most of the commonly used inferential techniques will be covered in detail.

 

Required Textbook:         Introduction to the Theory of Statistics, 3^rd ed. [MGB]

                                                   Mood AM, Graybill FA, Boes DC. Mcraw-Hill, 1974

Additional References: 1. Statistical Inference

                                                       Casella G, Berger RL. Wadsworth & Brooks, 1990 [CB]

2. Mathematical Statistics: Basic Ideas and Selected Topics

                                                        Bickel PJ, Doksum KA. Holden-Day, 1977 [BD]

Grading: Assignments (50%); Mid-Term Exam (30%); Final Exam (20%)

 

 

Brief Course Outline:

Part I: Theory of Distributions

I.1: Basic Probability Theory

I.2: Random Variables, Distribution Functions and Expectation

I.3: Common Parametric Families of Distributions

I.4: Joint Distributions

I.5: Distributions of Functions of Random Variable

 

 

Part II: Statistical Inference

II.1: Sampling Distributions

II.2: Parametric Point Estimation

II.3: Interval Estimation

II.4: Tests of Hypotheses

 

DETAILED COURSE OUTLINE: Part I

Week	Topics	Book Chapters: Sections	Assignments/Exams
1	Kinds of probability: classical, frequentist and axiomatic; Basic set theory: introduction	Chapter 1: Sec 1.1, 1.2.1-1.2.3,
	Sample space and event; finite sample spaces; conditional probability and independence	Chapter 1: Sec. 1.3.3-1.3.6	Assignment 1: (out) [MGB: Chapter 1]
2	Random variables and cumulative distribution function: Introduction and definitions	Chapter 2: Sec. 2.1, 2.2.1-2.2.2
	Discrete random variables; Continuous random variables; Expectations and moments	Chapter 2: Sec. 2.3.1-2.3.3, 2.4.1-2.4.2	Assignment 2 (out) [MGB: Chapter 2]
3	Chebyshev’s inequality; Jensen inequality; Moment generating functions	Chapter 2: Sec. 2.4.3-2.4.6
	Discrete distributions: Bernoulli, Binomial, Hypergeometric; Poisson	Chapter 3: Sec. 3.1, 3.2.1-3.2.2-3.2.4	Assignment 1 (due) Assignment 3 (out) [MGB: Chapter 3]
4	Discrete distributions: Geometric, Negative binomial; Continuous distributions: Uniform, Normal	Chapter 3: Sec. 3.2.5, 3.3.1-3.3.2
	Continuous distributions: Exponential, Gamma, Beta; Approximations; Poisson and exponential relationships	Chapter 3: Sec. 3.3.3-3.3.4, 3.4.1,3.4.2	Assignment 2 (due)
5	Joint distribution functions: Discrete and continuous, cumulative distribution function	Chapter 4: Sec. 4.1, 4.2.1-4.2.3
	Conditional distributions: Discrete and continuous, stochastic independence	Chapter 4: Sec. 4.3.1-4.3.4	Assignment 3 (due) Assignment 4 (out) [MGB: Chapter 4]
6	Conditional expectations, Joint moment generating functions; Independence and expectation; the Cauchy Schwartz inequality; Bivariate normal distribution	Chapter 4: Sec. 4.4.1-4.4.6, 4.5.1-4.5.3
	Distributions of functions of random variables: sum, product, quotient of random variables; The Cumulative distribution technique	Chapter 5: Sec. 5.1, 5.2.1-5.2.3, 5.3.1-5.3.4
7	Distributions of functions of random variables: The moment generating function technique, The transformation technique	Chapter 5: 5.4.1-5.4.2, 5.5.1-5.5.2, 5.6.1-5.6.2
	Sampling and sampling distributions: Definitions and basics; Law of large numbers; Central limit theorem	Chapter 6: 6.1, 6.2.1-6.2.4, 6.3.1-6.3.3	Assignment 4 (due) Assignment 5 (out) [MGB: Chapter 5]
8	Sampling and sampling distributions: normal distribution; The F distribution; The t distribution	Chapter 6: 6.4.1-6.4.5	Mid Term Exam

 

DETAILED COURSE OUTLINE: Part II

Week	Topics	Book Chapters: Sections	Assignments/Exams
9	Point estimation: Basics; Methods of finding estimators: Method of moments, maximum likelihood, other methods	Chapter 7: Sec 7.1, 7.2.1-7.2.3,
	Properties of estimators: closeness, mean squared error, consistency	Chapter 7: Sec. 7.3.1-7.3.3	Assignment 5 (due) Assignment 6 (out) [MGB: Chapter 7]
10	Properties of estimators (cont.): sufficiency , Factorization theorem, Minimal sufficiency	Chapter 7: Sec. 7.4.1-7.4.4, 7.5.1
	Properties of estimators (cont.): exponential family of densities, Unbiasedness (UMVUE), Location and scale invariance	Chapter 7: Sec. 7.6.1-7.6.2
11	Properties of estimators (cont.): optimal properties of maximum likelihood estimation	Chapter 7: Sec. 7.9
	Interval estimation: Definitions, Pivotal quantity, Sampling from the normal distribution	Chapter 8: Sec. 8.1-8.2, 8.3.1-8.3.4	Assignment 6 (due) Assignment 7 (out) [MGB: Chapter 8]
12	Interval estimation (cont.): Methods of finding confidence intervals (pivotal quantity method, statistical method)	Chapter 8: Sec. 8.4.1-8.4.2
	Interval estimation (cont.): large sample methods	Chapter 8: Sec. 8.5-8.6
13	Tests of hypotheses: Basics, definitions, simple vs. simple hypotheses	Chapter 9: Sec. 9.1-9.2.2
	Tests of hypotheses (cont.): composite hypotheses	Chapter 9: Sec. 9.3.1-9.3.4	Assignment 7 (due) Assignment 8 (out) [MGB: Chapter 9]
14	Tests of hypotheses (cont.): Normal distribution (on mean, variance, several means and several variances)	Chapter 9: Sec. 9.4.1-9.4.4
	Tests of hypotheses (cont.): Chi-square tests, contingency tables, goodness of fit	Chapter 9: Sec. 9.5.1-9.5.2
15	Tests of hypotheses (cont.): and confidence intervals ; Sequential tests (introduction)	Chapter 9: 9.6, 9.7.1
	General review of course		Assignment 8 (due) Final Exam