Question bank

All 347 questions, grouped by topic. 300 have a worked answer, folded away at the bottom of the question page. Or try a random question.

Probability

Random experiments and probability

Question	Answer
Three coin tosses	✓
Weights on the two coin spaces	✓
Drawing from 365 balls	✓
A lower bound on an intersection	✓
Fifty-four distinct birthdays	✓
Matching faces	✓
Three cards in order	✓
A point drawn from the unit interval	✓
Three people, one birthday	✓
One of each court card	✓
Three brothers in a random split	✓
Binary vectors with five ones	✓
Five balls out of twenty	✓
A one received on a noisy channel	✓
A zero received	✓
Information about one die	✓
A channel with unequal inputs	✓

Random variables and distributions

Question	Answer
The smaller face	✓
Ten voters	✓
The larger coordinate	✓
A cubic distribution function	✓
Twenty fair tosses	✓
Waiting for double six	✓
Ten balls in three boxes	✓
The sum of two dice	✓
The waiting time itself	✓
Distance to the centre	✓
Five draws with replacement	✓
Five draws without replacement	✓
Particles in an hour	✓
Five years of an exponential lifetime	✓
Three-point expectation	✓
Recovering the second moment	✓
Expected number of red balls	✓
Variance on the unit interval	✓
An affine image of a uniform variable	✓
Expected wait for double six	✓
Groups of three	✓
A memoryless component	✓
Why a normal variance is \sigma^2	✓
Exponential moments from the transform	✓
A game on the sum of two dice	✓
Two components with the same mean life	✓
A game on the smaller face	✓

Simulation

Question	Answer
Draws from an exponential law	✓
Draws from a uniform interval	✓
Rescaling a standard normal draw	✓
A shifted exponential density	✓

Joint distributions

Question	Answer
Minimum and average of uniform draws	✓
A joint table	✓
A probability as an integral	✓
Distribution function of a maximum	✓
One weight, or six weights	✓
A product density	✓
A point in a triangle	✓
Signs and their sum	✓

Functions of random variables and limit theorems

Question	Answer
The square of a binomial count	✓
The square of a uniform variable	✓
An affine image of a standard normal	✓
Two standard normal probabilities	✓
Probabilities for a shifted normal	✓
An affine image of an exponential density	✓
A normal approximation on an interval	✓
Sum and proportion of two exponentials	✓
The mean of repeated voltage readings	✓
A normal approximation in the tail	✓
A quadratic function of a uniform variable	✓
An affine image of a uniform variable	✓
A normal law with mean 2	✓
State the probability functions	✓
Partial sums of standard normals	✓
A lift and a weight limit	✓
Thickness of a circuit board	✓
The sum of uniform draws	✓
Paying the square of the number of heads	✓
A sieve for blueberries	✓
Ten successive exponential components	✓

Markov chains

Question	Answer
Two steps of a three-state chain	✓
The limit of the three-state chain	✓
Any two-state chain	✓
A chain that never settles	✓
A walk with a floor	✓
Where a random walk can be	✓

Reliability

Question	Answer
Three components, one of them essential	✓
Structure of a k-out-of-n system	✓
Pivoting the bridge	✓
Paths and cuts of the bridge	✓
A string of twenty bulbs	✓
Mean life as an integral of survival	✓
Survival from the hazard	✓

Statistical inference

Question	Answer
Summaries of ten cartons	✓
Which of these are statistics?	✓
The expectation of the sample variance	✓
A squared t variable	✓

Estimation

Question	Answer
Normal maximum likelihood estimates	✓
Bias of the two variance estimators	✓
An exponential rate	✓
A Poisson mean	✓
A shrunken sample mean	✓

Confidence intervals

Question	Answer
A proportion in a poll	✓
The interval for a normal mean	✓
Two standard errors	✓
Lost hours after a safety programme	✓
Checking equal variances	✓
Sizing a poll	✓

Hypothesis tests

Question	Answer
A critical region for a known variance	✓
Power of a serve test	✓
How small a variance?	✓
Paired or two-sample?	✓
Tyre wear on five cars	✓
A one-sided sign test	✓

Chi-squared tests

Question	Answer
A 1:2:1 phenotype ratio	✓
Is the die fair?	✓
A two-by-two table is a two-proportion test	✓
Counting degrees of freedom	✓

Linear regression

Question	Answer
Normal equations for a straight line	✓
A line through the origin	✓
How much do the ads explain?	✓
Where is the prediction interval narrowest?	✓
The correlation test is the slope test	✓

Analysis of variance

Question	Answer
Heart rates in four age groups	✓
A Tukey interval for two varieties	✓
Tips and coupons	✓
Two treatments in blocks	✓
A contrast among the corn varieties	✓
Two groups	✓

Data analysis

Studies, data, and evidence

Question	Answer
Alice’s heart-rate changes	✓
Randomising twenty people	✓
What a randomised trial does not need to record	✓
Facial tumours in Tasmanian devils	✓
A dose comparison for middle-of-the-night insomnia	✓
Alcohol and school attendance	✓

Descriptive statistics

Question	Answer
Classifying variables in a nutrition survey	✓
Reading a correlation	✓

Estimation examples

Question	Answer
Ten normal observations	✓
Household spending on fuel	✓
Narrowing a pilot interval	✓
Printer paper	✓
Hours of sleep	✓
A difference of proportions	✓
Variance of a fill volume	✓

Testing examples

Question	Answer
Caffeine in a cola	✓
A later poll on leaving	✓
Steel rod diameters	✓
A shuffling machine	✓
Two proportions	✓
Home-ground wins	✓

Analysis of variance examples

Question	Answer
Haemoglobin and sickle-cell type	✓
Five treatments, ten pairs	✓

Regression examples

Question	Answer
Height and age of pine trees	✓
Birth weight and mother’s age	✓
Rain, food, and orange yield	✓

Linear model examples

Question	Answer
How far sharks swim	✓
Wages, class, and gender	✓

Chi-squared examples

Question	Answer
Possums in five areas	✓
Genotype and mating success	✓

Bayesian statistics

Bayesian foundations

Question	Answer
Confidence procedures	–
What makes a hypothesis test?	–
One thousand posterior draws	–
Prediction is not estimation	–
The missing variance	–
Posterior mean under squared-error loss	✓
Q3. Confidence and credible intervals	✓
Q8. Equal-tailed and highest-density intervals	✓
Q13. Bayesian German tank problem	✓
Q15. Prior beliefs and conflicting data	✓
Q16. Interpreting identical interval endpoints	✓
C1. Likelihood, prior and posterior	✓
C5. Posterior prediction	✓
C6. Two kinds of interval	✓
C7. MAP and squared-error loss	✓

The Beta–Binomial model

Question	Answer
The pseudo-count trap	–
Symmetry and context	–
A proper nonconjugate prior	–
Sequential updating	–
A future run of tosses	–
A geometric observation and a uniform prior	✓
A logit-normal prior for a coin	✓
Q6. Beta-Binomial conjugacy and prior influence	✓
Q17. Choosing a conjugate prior	✓
Q42. A nonconjugate prior and a change of variable	✓
C2. Conjugacy	✓

Conjugate analysis for normal models

Question	Answer
Four limiting checks	–
Where did the probability go?	–
Find the prior rather than recalling it	–
How many summaries?	–
Full conditionals	–
Normal–inverse-\chi^2 conjugacy	✓
Q7. Normal estimation and prediction	✓
Q20. Estimation and prediction	✓
Q28. Lognormal variance and conjugate inference	✓
Q43. Unknown Normal mean and variance: conditional versus marginal	✓
Q49. A Gibbs full conditional under independent priors	✓
Q58. Joint Normal conjugacy and exact sampling	✓
Question 1. Normal mean, known variance	✓
Question 2. Normal variance, known mean	✓
Question 3. Normal mean and variance	✓
Question 4. Jointly conjugate normal–inverse-gamma	✓
Question 7. Lognormal variance	✓
Question 12. Improper prior for a normal sample	✓
C10. Gamma and inverse-Gamma kernels	✓

Markov chains, Gibbs sampling, and Metropolis–Hastings

Question	Answer
Detailed balance	–
A long run and a missed mode	–
Posterior summaries with dependence	–
Random-walk scale	–
Adapt, discard, fix	–
A bivariate-normal Gibbs sampler	✓
Metropolis–Hastings for the logit-normal posterior	✓
Random-walk Metropolis for a correlated normal	✓
An independent Gamma proposal	✓
Q4. Gibbs sampling from full conditionals	✓
Q5. Random-walk Metropolis-Hastings	✓
Q9. Gibbs sampling for a correlated normal target	✓
Q10. Metropolis-Hastings with an asymmetric-proposal extension	✓
Q14. Marginal and conditional distributions	✓
Q18. Equilibrium and independence	✓
Q19. Assessing a Metropolis–Hastings sampler	✓
Q25. Gibbs, Metropolis–Hastings, and Monte Carlo error	✓
Q26. General Beta–Binomial Gibbs sampler	✓
Q50. Metropolis–Hastings with a Cauchy prior	✓
Q51. Three-variable Gibbs sampling	✓
Q52. Bayesian logistic regression	✓
Q59. Gibbs sampling with a weighted count	✓
Question 6. Geometric–beta Gibbs sampler	✓
Question 9. Gibbs and random-walk Metropolis–Hastings	✓
Question 13. Binomial–beta Gibbs sampler	✓
Question 16. Mixed binomial–beta Gibbs sampler	✓
C11. Monte Carlo estimation	✓
C12. Markov property and invariance	✓
C13. Full conditionals and Gibbs	✓
C14. Metropolis–Hastings cancellations	✓
C15. Warm-up and dependence	✓

Categorical data and the Dirichlet distribution

Question	Answer
Beta hidden inside Dirichlet	–
Why not a multinomial prior?	–
Small concentrations	–
The simplex CLT	–
Dirichlet updating	–
Restaurant ordering methods	✓
Q22. Advertising responses and Dirichlet inference	✓

Bayesian hypotheses and model comparison

Question	Answer
Who owns the boundary?	–
Fruit posterior	–
Probability is not action	–
Make the simplex integral equal one	–
Two routes to the same table	–
An IQ threshold	✓
Q11. Bayes factors and posterior odds	✓
Q21. Evidence and posterior model probabilities	✓
Q29. Comparing Exponential and Gamma lifetime models	✓
Q44. Testing whether a coin is fair	✓
Q53. Poisson flaws and posterior odds	✓
Question 8. Exponential versus gamma lifetimes	✓
Question 15. Battery lifetimes and a Bayes factor	✓
Question 18. Bayesian model selection	✓
C8. Evidence and model odds	✓

Exponential families and conjugacy

Question	Answer
Factorisation and data reduction	–
A normal statistic	–
Reading conjugacy from a kernel	–
Three different “flat” Poisson priors	–
Conjugacy is not guaranteed by a name	–
Sufficiency and the posterior	✓
Exponential observations parameterised by rate	✓
Exponential observations parameterised by mean	✓
Q1. Rayleigh conjugate prior	✓
Q2. Gamma waiting times with an inverse-Gamma prior	✓
Q12. Pareto likelihood and Gamma posterior	✓
Q27. Machine downtime and a Gamma prior	✓
Q48. Conjugate Bayesian inference for a Gamma rate	✓
Question 10. Beta downtime	✓
Question 11. Geometric success probability	✓
Question 14. Inverse-gamma waiting time	✓
C3. Sufficiency	✓
C4. Exponential family	✓

Hierarchical Bayes

Question	Answer
Pooling	–
Derive the shrinkage weight	–
Exchangeability is a scientific claim	–
Two hyperpriors	–
Sample from a grid	–
Full Bayes and interval width	–
Q23. Rat-tumour studies and hierarchical shrinkage	✓
Question 17. Hierarchical Poisson model	✓
C9. Pooling	✓

Bayesian linear regression

Question	Answer
Errors and residuals	–
Normal before Student	–
The inverse-Gamma shape trap	–
Direct sampling or Gibbs?	–
Mean response or new response?	–
Independent and scaled priors	–
Q24. Bayesian regression and prediction	✓
Question 5. Normal–inverse-\chi^2 regression	✓

Generalised linear models

Introduction to generalised linear models

Question	Answer
Q33. Score, Hessian, and information identity	✓
Q35. Logistic regression and an adjusted effect	✓
Q56. Exponential regression and the score	✓
Q57. Negative-binomial canonical link	✓
C16. The GLM components	✓
C17. Links	✓
C18. Score and information	✓
C19. Fisher scoring and IRLS	✓
C20. Logit, probit and odds	✓

Standard errors, and models for counts

Question	Answer
Q30. Poisson regression model statement	✓
Q31. Interpreting Poisson regression coefficients	✓
Q32. Deriving the Poisson log-likelihood	✓
Q54. Poisson regression and overdispersion	✓
Question 19. Poisson regression for fish species	✓
C21. A log-link coefficient	✓
C22. Negative-binomial variation	✓

Underdispersion, positive responses, and quasi-likelihood

Question	Answer
Q45. Gamma regression from the exponential-family form	✓
Q47. Overdispersion and quasi-Poisson inference	✓
Q55. Gamma GLM for reaction times	✓
C23. Positive continuous responses	✓
C27. Quasilikelihood	✓
C28. Dispersion	✓

Categorical responses

Question	Answer
Q46. Multinomial logistic regression	✓
C24. Categorical response types	✓
C25. Proportional odds	✓

Paired comparisons, and how far a fit is from the data

Question	Answer
Q34. Deviance and model adequacy	✓
C26. Bradley–Terry	✓
C29. Saturated model and deviance	✓
C30. Significance is not adequacy	✓

Smoothing

Smooth means

Question	Answer
Q36. A binomial model with a smooth effect	✓
Q38. Changing from a logit to a probit link	✓
Question 21. Cholesterol and heart disease	✓
C31. Cubic splines and knots	✓
C32. Roughness penalty	✓

Additive smooths

Question	Answer
Q37. Testing and simplifying an additive model	✓
C33. GAMs and effective degrees of freedom	✓

Correlated data

Random intercepts

Question	Answer
C34. Fixed and random effects	✓

Random slopes, and a binary mixed model

Question	Answer
Q39. A generalised linear mixed model	✓
Q40. Interpreting a treatment–time interaction	✓
Q41. Random slopes and changed conclusions	✓
Question 20. Toenail infection GLMM	✓
Question 22. Orthodontic growth curves	✓
C35. Random intercepts and slopes	✓

Marginal models, and two responses at once

Question	Answer
C36. Conditional versus marginal dependence	✓
C37. Conditional versus marginal effects	✓
C38. Generalised estimating equations	✓
C39. Working correlation	✓
C40. Sandwich robustness	✓
C41. Vector-response regression	✓

Time series

Dependence along a series

Question	Answer
C42. Process and realisation	✓
C43. Two stationarity definitions	✓
C44. White noise is not necessarily independent	✓
C45. ACF and PACF	✓
C46. AR, MA and ARMA orders	✓
C48. Invertibility	✓
C50. ARCH and GARCH	✓

Forecasting

Question	Answer
C47. Unit roots and ARIMA	✓
C49. Identification and residual checks	✓