Math 50: Probability and Statistical Inference - WINTER 2006

Alex Barnett. Bradley room 308, tel 6-3178, email: m50w06 at

Bayesian solution of an inverse problem (x is parameters, y data)

Uncertainty governs both the data analysis done by scientists, and judgments made by us all in our everyday lives. In this course we apply the mathematical techniques of (mainly continuous) probability to estimation and hypothesis testing, the formal methods by which we learn from noisy data, random samples, and other such uncertain real-world measurements. We culminate with linear regression, and introduce the powerful framework of Bayesian inference.

These days, computers enable accurate analysis and visualization of probability models, and this course has a small but key computer component (using Matlab, R, or your favorite package). As well as improving understanding, you will learn valuable tools that have recently become the bread-and-butter of science (including social), economics, and medicine.

Jump to... Schedule, Resources, or Projects

Lectures / OH: Bradley 104, MWF 12:30-1:45pm (period 12). X-hr is 1-2pm Tues, and I imagine will be used about half the time (I will give a few days notice) for: quizzes, computer help, catch-up, or review material. Do not schedule anything regular in this X-hr. Office hours are 2-3 M, 4-5 Tu, and 2-3 F

Homework: 8-9 weekly HW's due Wednesday at start of lecture. I strongly encourage you to attempt the relevant homework problems before the next lecture. Leaving it all for Tuesday night is bad time management and risks you getting left behind in this fast-paced course. Please make your working/reasoning as clear as you can, write clearly, don't be scared of using lots of space on the page, and staple your work. Late homework will not be accepted (unless by prior arrangement for a valid, and exceptional, reason). Your lowest HW score will be dropped.

Exams: Two non-cumulative closed-book midterms (these will avoid the usual midterm season), and one final exam (single sheet of notes allowed). No algebraic/graphing calculators.

Real-world Statistics: Each week you should dig up a statistical example from the media, web, or other real-world source, post a paragraph to our comments page and be ready to explain it in class (I will pick on you, randomly of course!). If it relates to the week's content, all the better. As the course progresses we will be able to connect these to the material. Why do it?

  1. statistics is all around us, affects policies, our lives, etc. The discipline was invented to deal with these questions.
  2. communication skills
  3. project ideas for you and all of us
  4. choose what interests you
Project: I am keen to have you do a project in the last 1-2 weeks, worth at least 10%, in which you apply what you've learned to analyse real-world statistical data. Stay tuned.

Honor principle. Exams: no help given or received. Homework/Project: no copying, however collaboration on problem techniques is encouraged. Write-ups must be done individually.

Grades: Your overall grade will be computed according to HW 15%, Midterms 2*20%, Final+Project 40%, Real-world-statistics contributions 5%. Note that although HW has a low weighting, it is the main chance you get to practise the material and get feedback. Grades in Math 50 are not curved; other students' good performance will not hurt your grade. (So please work together and help each other out.)


weekdatereading (LM4)homework (due following Wed) / daily or weekly topics / info
1Jan 4 W website, Ch.1, 2.1-3HW1. Overview, learning from data (frequentist vs Bayesian) Bayesian coin applet
6 F2.4-5review conditional, independence.
7 Saspecial preemptive catch-up day! (free)
9 M 2.6-7, 3.2Combinatorics, hypergeometric
10 Tu X-hr3.3-4(lecture to replace Sat) Binomial, random variables, PDF, cumulative distn func.
211 W3.5HW2. Expectation values, including binomial and hypergeometric. [2-4pm Susan A. Schwarz's MATLAB intro tutorial]
13 F3.6variance
16 M(MLK holiday: no class)
17 Tu X-hr3.7(lecture to replace MLK class) joint densities, marginals
318 W3.8-9HW3. Combining variables (convolution 1, 2), worksheet solutions, mean and variance of such.
20 F3.10-11 order statistics, conditional pdfs.
23 M4.2 Poisson distribution (poisson.m code, process, bus paradox 1, 2)
24 Tu X-hr(free)
425 W4.3HW4. Normal (gaussian) distribution (standard normal cdf applet).
27 FMidterm 1: on material from HW 1-3 (solutions, practise qu's).
29 M 4.4-5 geometric and negative binomial pdfs
5Feb 1 W 4.6, 5.1-2 HW5. Gamma pdf and function. Estimation: max likelihood (lik.m 1-param max likelihood demo).
3 F 5.3-4 (lik_gamma2.m 2-param gamma likelihood plot) confidence intervals
6 M 5.5 properties of estimators: bias (applet demo: ML variance estimator is biased).
7 Tu X-hr5.6 efficiency, minimum-variance estimators (lecture to replace Carnival class).
68 W 5.7-8 HW6 (selected answers, code for qu C). Cramer-Rao lower bound, consistency (worksheet solutions).
10 F(Carnival holiday: no class)
13 M5.8, 6.2 Bayesian estimation. Hypothesis testing.
14 Tu X-hr Problem-solving session (optional).
715 W 6.3 HW7 (solutions). Binomial hyp testing.
17 F 6.4 Type I, II errors.
20 MMidterm 2: on material from HW 4-6. (solutions, practise qu's with answers).
21 Tu X-hr7.2Normal distribution, t test (student.m t-distn demo). Choose projects, start work on them.
822 W7.4 inference on mean of normal data, t tests
24 FBayesian inference for (mu, sigma) of normal data, nonlinear transformation of pdfs.
27 M7.5Random sampling from any pdf, inference on variance of normal data. reading data into matlab. chi.m, chi-squared demo
28 Tu X-hr (free)
9Mar 1 W9.2, 4Two-sample tests: means (t-test w/ equal-variance), and proportions (normal approximation).
3 F11.4Covariance and correlation coefficient (corr_regr.m demo).
6 M11.2, 5 Linear regression, robust noise models, Bayesian model fitting with Markov-chain Monte Carlo (MCMC). (regr_mcmc_sampling.m demo, needs nlp_linear_gauss.m, nlp_linear_exp.m, mcmc_run.m).
7 Tu X-hr (free)
108 W (last day of class) Class project presentations (continues w/ pizza, 6:30-8pm, Bradley 105).
10 F Review session (usual time 12:30-1:35). Post-Mid2 practise problems w/ solutions. Project reports due (midnight)
13 MFinal Exam (solutions): March 13th at 8:00 am - 11:00 am, Bradley 104 (usual room). Note this exam will not be given early to accommodate travel plans.

Special needs: I encourage students with disabilities, including "invisible" disabilities like chronic diseases and learning disabilities, to discuss with us any appropriate accommodations that might be helpful. Let me know asap, certainly in first 2 weeks. Also stop by the Academic Skills Center in 301 Collis to register for support services.

Private tutoring: Tutor Clearinghouse may have private one-on-one tutors available for Math 50. The tutors are recruited on the basis that they have done well in the subject, and are trained by the Academic Skills Center. If a student receives financial aid, the College will pay for three hours of tutoring per week. If you would like to have a tutor, please go to 301 Collis and fill out an application as early in the term as possible.