## BIOS 6341: Introduction to Probability and Statistical Theory (Fall 2016) See BIOS6341Course for the course's current syllabus.

### Schedule

• Lectures: Monday, Wednesday, and Friday, 10:00-11:00, Biostat Conference Room

• Lab: Friday 11:00-12:00, Biostat Conference Room

• Office hours:

• Bryan: 3-4pm Mondays, 2525 West End #11124
• Rui: 3-5pm Thursdays, Garden Room (#11101A)

### Other information

• Our textbook is Statistical Inference, Second Edition by Casella G and Berger RL.
• The tentative lecture schedule is shown in the table below.
• Students are expected to read the noted section in the text prior to each class.
• Homework is due at the beginning of class on the date noted.
• Students are encouraged to work together on homework problems, but they must turn in their own write-ups.
• Generally, you should bring a laptop to the lab on Fridays

• Homework / Class Participation: 20%
• Midterm Exams: 40% (20% each)
• Final Exam: 40% I expect the final exam will be 3 hours, closed book. One third of the material will be from Chapter 5 and the remaining two-thirds from all chapters (chapters 1-5).

### Lectures (tentative)

Date Lecture Topic Reading Homework Due at Start of Class
Aug 23 1 Introduction and Set Theory CB 1.1
Aug 25 2 Axiomatic Foundations / Calculus of Probabilities 1.2 Homework for Lectures 1-2
Aug 25 Lab 1 Poker Probabilities ( poker-draw.R)
Aug 28 3 Counting / Enumerating Outcomes 1.2 1.16, 1.17, 1.18, 1.20, 1.22
Aug 30 4 Conditional Probability and Independence 1.3 1.33, 1.34, 1.36, 1.37b, 1.38, 1.39, 1.40
Sept 1   Review and Discuss Homework problems     HW for lectures 1-4
Sept 1 Lab 2 Birthday problem ( b-day.R)
Sept 4   LABOR DAY
Sept 6 5 Random Variables /Distribution Functions 1.4-1.5
Sept 8 6 Density and Mass Functions 1.6 1.49, 1.50, 1.51, 1.53, 1.54, 1.55
Sept 8 Lab 3 Delirium Study, Conditional Probability, and Causal inference; delirium.pdf
Sept 11 7 Distributions of Functions of a Random Variable 2.1 2.1, 2.2, 2.3, 2.4, 2.6 (don't need to show pdf integrates to 1), 2.8 (don't need to show it's a cdf)
Sept 13 8 Expected Values 2.2
Sept 15   Review and Discuss Homework Problems     HW for lectures 5-7
Sept 15 Lab 4 Distributions and Transformations ( distributions-transformations.R)   Generate 1000 X from GAM(3,2) distribution. Compare empirical density and cdf with true density and cdf. Generate Y=UNIF(0,1) using probability integral transformation and verify that empirical cdf is similar to true cdf. Next, generate 1000 U from UNIF(0,1) and then generate V~GAM(3,2) using the inverse cdf of a gamma distribution. Please turn in plots of your simulations as well as printout of your code.
Sept 18 9 Moments and Moment Generating Functions 2.3 2.15, 2.20, 2.24, 2.33. EXTRA: Let X be a non-negative continuous random variable with CDF F(x) and E(X)< infinite. Show that E(X)=integral from 0 to infinity (1-F(x))dx.
Sept 20 10 Discrete Distributions 3.1-3.2 3.1, 3.2, 3.3, 3.4, 3.5 (probably need to use R), 3.7, 3.8
Sept 22   Review and Discuss Homework Problems     HW for lectures 8-9, and Lab 4.
Sept 22 Lab 5 Review for Exam ( 2011 Exam and Solutions, 2014 exam, 2014 solutions)
Sept 25   EXAM
Sept 27 11 Continuous Distributions 3.3 3.17, 3.22d, 3.23, 3.24a (hint: substitute z=y^gamma/beta),3.24c (hint: substitute z=1/y), 3.25, 3.26
Sept 29   Review and Discuss Homework Problems     HW for lecture 10
Sept 29 Lab 6 Review and Discuss Exam
Oct 2 12 Exponential Families / Location and Scale Families 3.4-3.5 3.28 (for a-c do it only for both unknown), 3.29, 3.30 (some versions of the book have part b for the beta distribution-- don't do this; part b should be for a Poisson distribution), 3.37, 3.42
Oct 4 13 Joint and Marginal Distributions 4.1 4.1, 4.1d: P( abs(X+Y) <1), 4.4, 4.5
Oct 6 14 Review and Discuss Homework Problems     HW for lecture 11-12
Oct 6 Lab 7 Survival Analysis (Exponential Distributions and Censoring)
Oct 9 14 Conditional Distributions and Independence 4.2 4.7, 4.9, 4.10, 4.11, 4.12, 4.13
Oct 11 15

Bivariate Transformations

[BRYAN OUT]

4.3 4.15, 4.16, 4.19, 4.20, 4.22
Oct 13   FALL BREAK
Oct 16 16

Hierarchical Models and Mixture Distributions

[BRYAN OUT]

4.4 4.31, 4.32a, 4.34a, 4.35
Oct 18 17 Covariance and Correlation 4.5 4.41, 4.42, 4.43, 4.45a-b, 4.58a-b
Oct 20   Review and Discuss Homework Problems     HW for lectures 13-16
Oct 20 Lab 8
Oct 23 18 Multivariate Distributions 4.6

4.36, 4.39 (hint for Cov(X1+X2): find Var(X1+X2)),

Using pdf in Example 4.6.1, find a) f(x1,x2,x3), b) f(x4 given x1,x2,x3), c) P(X1<1/2,X2<1/2,X3<1/2), d) P(X4<1/2 given X1=X2=X3=1/2).

Oct 25 19 Inequalities and Identities 3.6 and 4.7 3.46, 4.63
Oct 27   Review and Discuss Homework Problems
HW for lectures 17-19
Oct 27 Lab 9 Review for EXAM ( 2011 exam with solutions, 2014 exam , solutions-a, solutions-b, solutions-c)
Oct 30   EXAM
Nov 1 20 Random Samples and Sums of Random Variables 5.1-5.2 5.1, 5.3, 5.5, 5.8a, c (assume E(X)=0 and use part a)
Nov 3   Review and Discuss EXAM
Nov 3 Lab 10 Ordinal Residual
Nov 6 21 Normal Distribution (Properties of Sample Mean and Variance) 5.3 5.10 (use Stein's lemma for a), 5.11, 5.15, Additional Problem: Xi ~ iid N(mu, sigma^2). a) show that Cov(X1-Xbar,Xbar)=0; b) use (a) to show that xbar is independent of S^2.
Nov 8 22 Normal Distribution (Derived Distributions) 5.3 5.17, 5.18a,b,c (use version of Sterling's formula given in 5.35b)
Nov 10   Review and Discuss Homework Problems     HW for lectures 20-21
Nov 11 Lab 11 Approaches for generating a random sample 5.6
Nov 13 23 Order Statistics 5.4 5.21, 5.22, 5.24, 5.27
Nov 15 24 Convergence Concepts (convergence in probability, a.s., distribution) 5.5 5.32, 5.42
Nov 17   Review and Discuss Homework Problems
HW for lectures 22-23
Nov 17 Lab 12 Approaches for generating a random sample R code 5.6
Nov 20   THANKSGIVING
Nov 22   THANKSGIVING
Nov 24   THANKSGIVING
Nov 27 25 Convergence Concepts (central limit theorem) 5.5 5.29, 5.30, 5.31, 5.34, 5.35
Nov 29 26 Convergence Concepts (delta method) 5.5 5.44. Additional problem: Let Xi ~ iid BIN(1,p1), Yi ~ iid BIN(1,p2), i=1,...n for both, all Yi independent of Xi; What is the limiting distribution of the sample log odds ratio (logOR) where logOR=log(p1(1-p2)/(p2(1-p1)))?
Dec 1   Review and Discuss Homework Problems
HW for lectures 24-25, and Lab 12
Dec 1 Lab 13 Approaches for generating a random sample 5.6 accept-reject.R: Code to generate from Beta(2,2) distribution with Unif(0,1) ; accept-reject-beta.R: Code to generate from Beta(6.1,1.8) distribution with Unif(0,1)
Dec 4 27

Review and Discuss Homework Problems

Code generating from biased Weibull distribution HW for lecture 26 and labs 13
Dec 6

Review for FINAL EXAM

Dec 11, 12-3   FINAL EXAM 2015 Exam and Solutions, 2014 Exam and Solutions, 2016 Exam and Solutions

Topic revision: r100 - 06 Aug 2019, RobertGreevy

• Biostatistics Webs Copyright © 2013-2020 by the contributing authors. All material on this collaboration platform is the property of the contributing authors.
Ideas, requests, problems regarding Vanderbilt Biostatistics Wiki? Send feedback