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Last edits are highlighted in yellow, added in accordance with this question and answer on Ed Discussion.

When and where

Our final exam scheduled time slot for this class is Tues Dec 17 at 2:00 pm. The assigned location is now SPL 59 (this was changed from the originally stated location). This information can be found at this page.

Note “SPL” is “Sloane Physics Laboratory”.

The official length of the exam is 3 hours (180 minutes), but you will be allowed to work on it for up to 3.5 hours (210 minutes).

[The odd form of the last statement comes from the following Yale policy as stated in The Yale College Programs of Study: “final examinations normally last either two or three hours but, in either case, students are permitted to take an additional half hour before being required to turn in their answers. This additional time is given for improving what has already been written, rather than for breaking new ground.”]

What to bring

  • Bring a calculator (as described in the “IMPORTANT CHANGE” Announcement sent from Canvas on Dec 10 at 3:54pm).
  • You can bring up to 4 sheets of notes, writing (or typing) whatever you would like on both sides of 4 letter-sized (\(8.5\times 11\) inch) sheets of paper.
    • Please do not bring a microscope or magnifying glass to read your notes! They should be written at a reasonable size (e.g. \(\ge 10\) pt font if typed) to be readable comfortably without optical assistance by people with reasonable visual acuity.
    • We’ve had just 2 sheets in prior years… 4 sheets is intended to be more than enough and reduce stress in creating the notes; I hope you don’t feel any need to write tiny or even fill 4 sheets.

About calculations

This is changed from the last version.

Unless otherwise specified: For all questions that have numerical answers, you should calculate explicit numbers, to at least 3 significant figures of accuracy. You can use the calculator you brought. You should use the basic standard numerical functions of arithmetic, trigonometry, powers, and exponentials (and not statistical, symbolic math, and graphing functions), and when writing up answers to get full credit the calculations should be written in a way that only relies on those basic calculator functions. Try to maintain enough decimals of accuracy throughout so that your final answers are accurate to at least 3 significant figures. For the normal cdf, use the included normal table, not calculators; round z-scores off to two decimal places to use the normal table, and don’t bother with continuity corrections.

If you are asked for R code, your goal is to write accurate code that a grader could easily type into an R session and see the desired answer pop out. Your code should be neatly written and easily read.

A couple of notes about R code: Even if asked to give your answer using R code, if it’s easier or more convenient (for example, for problems involving math), you are welcome to do your work using mathematical notation throughout or as much as you want, and only at the end write your answer using R code. If the question specifies the desired length of your answer as an R expression, for example, “a single line”, then you should give a single line of R code (without any semicolons that would allow combining several lines of code on one line). In other words, just do not use any semicolons in your R code.

Coverage

A common question is, “is the exam cumulative?” Sort of, no and yes. No, in the sense that the exam is intended to emphasize material after the midterm exam. But of course the subject itself is quite cumulative; for example, everything we do ultimately depends on basic probability, so material from the whole semester may be relevant to any given part of a problem, even though I would not include a problem on the final whose sole purpose is to test material from before the midterm.

Here are the topics I can recall as of this moment; I may have forgotten things and will try to add them as I think of them. I won’t repeat the list of topics that were covered by the midterm (if you want to look back at those you can find them here).

  1. Likelihood functions, maximum likelihood estimators
    1. Should be able to write down likelihood functions and find maximum likelihood estimators
  2. Prior, posterior, Bayes’ rule
  3. Law of total probability and law of total expectation
  4. Variance
    1. Definition, rules
    2. Of course you should know about means (expected values) also, but I’m not writing all the topics that are covered by the first midterm here (you can see this page).
  5. Law of Large Numbers
    1. Markov inequality
  6. Normal distributions, Central limit theorem, Normal approximations
    1. Normal pdf’s: standard and general, idea of exp(quadratic function)
    2. “68, 95, 99.7 rule”
    3. standardizing
    4. Use of standard normal table to calculate cumulative probabilities (standardize, round to two decimal places, look up in Normal table)
    5. Distributions of sums; convolutions
    6. Central limit theorem statement and use for Normal approximations
  7. Markov chains
    1. Basic concepts: state space, initial distribution, Markov transition matrix, Markov property
    2. Stationary distributions
    3. Detailed balance (time reversibility)
    4. Ergodic theorem
  8. Markov chain Monte Carlo
    1. General Metropolis-Hastings method, and special cases like random-walk Metropolis, independence sampler, and Gibbs sampler
    2. How to apply MCMC to Bayesian inference, including: prior, likelihood, posterior, generating proposals, accepting proposals, creating inferential summaries like histograms of posterior and credible intervals and posterior probabilities; how to do inference about a function of the parameters in the model
  9. Conjugate priors for Bayesian inference
    1. Idea of updating parameters from prior to posterior in a conjugate family, and the following examples.
    2. Beta prior for binomial success probability
    3. Normal prior for a Normal mean
    4. Gamma prior for a Normal precision
  10. Correlation and regression
    1. Covariance and its rules
    2. Definition of correlation
    3. Equation of the regression line
    4. “Regression effect”
    5. Calculations like those in pset 10 question #2 and section 5.6.4 of the course textbook, including conditional distributions of \(Y\) given \(X=x\).
    6. Multiple regression models
  11. Distributions including Binomial, Geometric, Uniform, Normal, Cauchy, Exponential, Beta, Gamma
  12. Other computing and R topics
    1. Vectors, indexing, extracting parts of vectors, matrices, rows, columns, matrix multiplication
    2. Assigning variables, defining functions, how functions work (e.g. arguments, default values)
    3. Loops
    4. Simulation (for example birthday problems)
    5. Some particular functions: arithmetic (+,-,*,/,^), <-, =, ==, exp, log, sqrt, choose, c, length, mean, sum, cumsum, prod, rep, seq, max, min, table, quantile, sample, hist, truehist, plot, abline, points, lines, rowSums, colSums, var, cor, convolve, which, sapply, as.data.frame, $ to refer to columns, read.csv
    6. Distributions: each family has 4 functions beginning with “d”, “q”, “q”, and “r”; you should know what these are. E.g. Normal family has dnorm(x,mu,sigma), pnorm(x,mu,sigma), qnorm(x,mu,sigma), rnorm(n,mu,sigma)
    7. Markov chains: calculating distribution at time n, simulating a sample path given the initial distribution and transition matrix
    8. R code for Metropolis sampling
    9. JAGS: how to write a model; two types of JAGS model statements; how the distributions we’ve been using are written in JAGS; priors that could be considered reasonable default choices; how to run the model using R functions jags.model, coda.samples; how to make the results into a data frame and create inferential summaries and other items above in “How to apply MCMC…” section; Deviance Information Criterion for model selection

Practice problems

Two sets of practice problems are linked to the course home page in Week 14. For each set of sample problems, one file has just the problems, and another has the problems with solutions.

Advice on studying

I would also advise going back over the previous homework assignments thoroughly and try to make sure you really understand them. I think it is good to re-do those problems from scratch, with only a blank sheet of paper in front of you, seeing if you can come up a solution without help. If you have difficulty, then you can take a peek at the solution or at your own pset, with your eyes squinted as if looking toward the sun, only looking for as brief a time as possible until you get any inkling of a hint that could help you make progress. Then go back to the sheet of paper in front of you and make some progress, and either finish the problem, or get stuck and take another minimal peek, and so on. After you have fully solved a question, then it is good to take one more thorough read through the solution to make sure you haven’t missed any tips or wisdom that might be hidden there. You want to get to the point where you have thoroughly understood each question and absorbed whatever lessons, techniques, and problem solving strategies you might be able to derive from them.

This same pattern can be applied to all practice problems. You can try these problems first, preferably under exam-like conditions, and then make sparing use of the solutions as necessary as described above, and then finally after completing the problem read through the solution carefully.