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---++ BIOS 6341: Introduction to Probability and Statistical Theory (Fall 2016)
---+++ Instructor
   * Bryan Shepherd, !PhD, Associate Professor of Biostatistics
   * bryan.shepherd@vanderbilt.edu; 2525 West End, 11124; 343-3496

---+++ Teaching Assistant
   * Rui Wang
   * rui.wang.1@vanderbilt.edu
---+++ Schedule
   * <p>Lectures: Monday, Wednesday, and Friday, 10:00-11:00, Biostat Conference Room</p>
   * <p>Lab: Friday 11:00-12:00, Biostat Conference Room</p>
   * <p>Office hours:</p>
      * 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

---+++ Grading (tentative)
   * 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 |  [[%ATTACHURL%/hw.pdf][Homework for Lectures 1-2]] | ||||||
| Aug 25 | Lab 1 | Poker Probabilities ( [[%ATTACHURL%/poker-draw.R][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 ( [[%ATTACHURL%/b-day.R][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; [[%ATTACHURL%/delirium.pdf][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 ( [[%ATTACHURL%/distributions-transformations.R][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)&lt; 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_ | | | <em>HW for lectures 8-9, and Lab 4.<br /></em> ||||||
| Sept 22 | Lab 5 | _Review for Exam_ ( [[%ATTACHURL%/exam1-2011-solutions.pdf][2011 Exam and Solutions]], [[%ATTACHURL%/exam1-2014.pdf][2014 exam]], [[%ATTACHURL%/20150925160152.pdf][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_ | | | <em>HW for lecture 10<br /></em> ||||||
| 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) &lt;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 | <p>Bivariate Transformations</p> <p>[BRYAN OUT]</p> | 4.3 | 4.15, 4.16, 4.19, 4.20, 4.22 | <i><br /></i> ||||||
| Oct 13 | | FALL BREAK | | | <em><br /></em> ||||||
| Oct 16 | 16 | <p>Hierarchical Models and Mixture Distributions</p> <p>[BRYAN OUT]</p> | 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 | <p>4.36, 4.39 (hint for Cov(X1+X2): find Var(X1+X2)),</p> <p>Using pdf in Example 4.6.1, find a) f(x1,x2,x3), b) f(x4 given x1,x2,x3), c) P(X1&lt;1/2,X2&lt;1/2,X3&lt;1/2), d) P(X4&lt;1/2 given X1=X2=X3=1/2).</p> | ||||||
| Oct 25 | 19 | Inequalities and Identities | 3.6 and 4.7 | 3.46, 4.63 | ||||||
| Oct 27 | | <em>Review and Discuss Homework Problems<br /></em> | | | _HW for lectures 17-19_ ||||||
| Oct 27 | Lab 9 | _Review for EXAM_ ( [[%ATTACHURL%/exam2-2011-solutions.pdf][2011 exam with solutions]], [[%ATTACHURL%/exam2-2014.pdf][2014 exam]] , [[%ATTACHURL%/20151030142927.pdf][solutions-a]], [[%ATTACHURL%/20151030142905.pdf][solutions-b]], [[%ATTACHURL%/20151030143330.pdf][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_ | | | <em>HW for lectures 20-21<br /></em> ||||||
| 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 | | <em>Review and Discuss Homework Problems<br /></em> | | | <em>HW for lectures 22-23<br /></em> ||||||
| Nov 17 | Lab 12 | Approaches for generating a random sample [[%ATTACHURL%/code-order-CLT.R][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 | | <em>Review and Discuss Homework Problems<br /></em> | | | <em>HW for lectures 24-25, and Lab 12<br /></em> ||||||
| Dec 1 | Lab 13 | Approaches for generating a random sample | 5.6 | [[%ATTACHURL%/accept-reject.R][accept-reject.R]]: Code to generate from Beta(2,2) distribution with Unif(0,1) ;  [[%ATTACHURL%/accept-reject-beta.R][accept-reject-beta.R]]: Code to generate from Beta(6.1,1.8) distribution with Unif(0,1)| ||||||
| Dec 4 | 27 | <p> _Review and Discuss Homework Problems_ </p> <p> </p> | | [[%ATTACHURL%/accept-reject-HIV.R][Code generating from biased Weibull distribution ]] | _HW for lecture 26 and labs 13_ ||||||
| Dec 6 | | <p> _Review for FINAL EXAM_ </p> <p> </p> | | | ||||||
| | | | | | ||||||
| Dec 11, 12-3  | | FINAL EXAM  |  [[%ATTACHURL%/20151215131626259.pdf][2015 Exam and Solutions]], [[%ATTACHURL%/20141210153400.pdf][2014 Exam and Solutions]], [[%ATTACHURL%/20161213133132052.pdf][2016 Exam and Solutions]] | | ||||||
Topic revision: r98 - 04 Dec 2017, BryanShepherd
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