Statistics

Three readings from a Pareto model with known minimum 1 are e, e² and e⁴. Find the maximum likelihood tail index.

For uniform data the likelihood ratio statistic is not a smooth function, and the rejection region has two separate pieces.

Observe 6 successes in 8 trials. Is a success probability of 1/3 plausible? Doubling the tail is not the only way to answer.

The mean and the standard deviation are tied to the same unknown number. What is the Cramér-Rao bound for estimating it from six observations when it equals 2?

Run the test at n, and if it fails, double the sample and test again at the same level. The Type I error is no longer 5%. What is it?

Compute the deviance-style statistic for 60 heads in 100 tosses. Natural logarithms, three decimals.

Draws come from Uniform(θ, θ+1). The likelihood is flat across a whole interval of θ values. How long is it?

In 100 individuals the genotypes AA, Aa and aa number 30, 50 and 20. Under random mating, what is the likelihood-best frequency of allele A?

In naturally occurring datasets, what fraction of numbers start with the digit 1?

A sample consists of 2, 4, and 6. What is the sample variance, and why divide by n − 1?

Six counts come from the same Binomial(N, p) with both N and p unknown. Find the maximum likelihood estimate of N.

Three strata with different spreads and interview costs: spend a fixed budget optimally and find the best possible variance.

Six bulbs are tested for 12 hours. Three burn out early; three are still lit when the test ends. Estimate their mean life.

For any distribution whatsoever, what fraction of observations must lie within 2 standard deviations of the mean?

A shrinkage estimator for a binomial proportion with a peculiar property, its mean squared error is the same at every true value. Evaluate it at p = 1/10 and see what is special.

X is uniform on [0,1]. How strongly are X and X squared correlated, despite one being a curved function of the other?

Given only variances and one covariance, find how strongly X+Y moves with X-Y.

Ten fair flips contain nine adjacent pairs, but the pairs overlap. Find the exact variance of the number of pairs that are both heads.

Two unbiased estimators of the upper endpoint of a uniform distribution, twice the sample mean or a rescaled maximum. How much noisier is the first?

A lake holds N fish. Tag 50, release, then catch 40, and eight are tagged. The likelihood is maximized not once but twice. Give the smaller maximizer.

Thirty tagged fish are in a lake. You keep fishing until the fourth tagged one appears, which takes 25 catches. Estimate the lake population.

Your sample of size n gives a standard error SE. How big a sample cuts the standard error in half?

Both methods control the family-wise error rate at 5%, but one of them quietly rejects more than twice as many hypotheses on this list.

Sampling whole classrooms is cheap but students in the same room resemble each other. How many rooms match a simple random sample of 600?

A trial wants 80% power to detect a modest difference. Plan the enrolment arithmetic before the first patient is recruited.

A success has probability 0.3 per trial, and you saw 5 successes but forgot the number of trials. Which count is most likely?

Every pair of n variables has correlation -3/20. How large can n be?

Five points, a least-squares line, and the fraction of the variance in y that the line accounts for.

Heavy tails make single observations nearly useless, yet every one carries a fixed amount of Fisher information. What is the best possible variance for ten of them?

A Poisson count is thrown away and only the question "was it zero?" is recorded. How much Fisher information about the rate survives?

Benford's law is not about the number ten. In base 16, how often does a number lead with a digit from 1 to 3?

Six observations, one fitted line, and an intercept that is not an integer. Exact arithmetic only.

Four values are drawn uniformly from [0, 1]. What is the expected value of the largest one?

You observe n draws from Uniform(0, θ) with θ unknown. What is the maximum likelihood estimator of θ?

Serial numbers are drawn from 1 to N. Equate the population mean to the sample mean. The estimate lands below an observed number.

A fair coin is flipped ten times, and the experimenter may declare victory whenever heads lead tails by four. What are the odds that the fair coin gets declared biased?

You can only record whether each interval was empty. Twenty of fifty were. What rate do you infer?

Reject when the largest of four uniform readings is small. How often does that happen when the true range is only a quarter shorter?

A test that rejects on at least 8 heads in 10 flips. What fraction of the time does it catch a coin that lands heads 75% of the time?

Praised cadets get worse; berated cadets improve. The instructor credits punishment. What is really going on?

The product of two coin flips is a crude unbiased estimate of p². Condition on the total number of heads and its variance collapses. By how much?

Three observation windows of different lengths record 3, 10 and 0 events. What single rate per hour does maximum likelihood give?

Seven measurements, two groups, no distributional assumptions. Count the relabelings that look at least as extreme.

Six A's and four B's land in random order. How likely is it that they cluster into three or fewer blocks?

The sample variance is unbiased only for independent data. For three autocorrelated observations, what does it actually estimate?

Rank the absolute differences, add up the positive ranks, and count sign patterns exactly.

A small clinical table, a hypergeometric null, and a two-sided p-value that is not simply twice the one-sided one.

Treatment A beats B on small stones AND on large stones. Can B still win overall when the groups are combined?

Five subgroups, five polls, and one promise. All five intervals must be right together with 95% confidence. How big must each poll be?

A fair coin is flipped 100 times. Roughly how likely are 60 or more heads?

Multiply by m over rank and you get 0.036, yet the correct adjusted p-value is smaller. Why, and what is it?

If the null is true and the data are continuous, p is uniform with mean 1/2. For four coin flips, what is the mean p-value?

A screen tests a thousand hypotheses at nominal FDR 5%. The true false discovery rate is not 5%. What is it?

Pick the manager who beat the market five years running. Is she skilled, or just the lucky survivor of a large population?

You capture four tanks with serial numbers 19, 40, 42, and 60. How many tanks does the enemy have?

An auditor sees two figures in a ledger, both ending in 0 or 5. How suspicious should that make her, given a modest prior of fraud?

WWII bombers returned riddled with holes on the wings but not the engines. Where did Abraham Wald say to put the armor?

A single observation, two candidate distributions, and a randomized test. Rank outcomes by likelihood ratio, not by size.

Two correlations of 0.8 are fixed. How low can the third one go before no random variables can exist?

Sample families uniformly and one-child families are common. Sample children uniformly and the odds shift. Compute the exact chance.

An analyst checks an A/B test's p-value daily and stops the first time p < 0.05. Does the 5% guarantee survive?

Four of five y-values are known. What must the fifth be to make the least-squares slope exactly zero?

No normality and no standard error, just the smallest and largest of a handful of observations. How often does that interval catch the median?

A 99%-accurate test for a disease affecting 1% of people comes back positive. What is the chance you actually have it?

Test scores correlate 0.6 between two sittings. Regression to the mean works backwards in time too, which surprises people.

Two independent dice, but only pairs summing to at least 10 are admitted. What is the correlation inside the admitted pool?

A discrete test has to stop short of the nominal 5% level. How short? Find the true size of a sign test.

A lab runs ten independent tests, finds one striking p-value, and reports it. How striking is it really once you account for the search?

One p-value in the middle of the sorted list fails its own threshold, yet it is still rejected. How many discoveries does the procedure report?

85% of cabs are Green, and a witness who is right 80% of the time says the cab was Blue. How likely was it actually Blue?

One fair coin, one two-headed coin. You pick one at random and flip heads three times. What is the chance it's the trick coin?

Buses average one per 10 minutes and you arrive at a random time. Why is your expected wait not 5 minutes?

Inside every group the trend is exactly downhill. Pool the data and the fitted slope turns uphill.

Four makes in five shots, a skeptical prior, and a question about the next three: how does conjugacy turn this into a product?

Dividing the sum of squared deviations by something larger than n-1 adds bias but removes variance. How much total error can you buy back?

Data come from Uniform(θ, 3θ). The method of moments and maximum likelihood disagree. By how much?

Twenty null effects, each tested at the 5% level. How likely is it that the data hand you at least two "discoveries"?

A Gamma prior on a Poisson rate, a little data, and a question about two future days with no events at all.

To estimate the chance that a Poisson count is zero, start with the crude indicator on the first observation and condition on the sufficient total.

The sample variance of three die rolls is unbiased, but how much does it bounce around? Compute its exact variance.

What is the variance of the sum of two independent fair dice?

Bus gaps are the sum of two exponential stages. Arrive at a random moment: how long, on average, until the next bus?

Is it correct to say "there is a 95% probability the true mean lies in this interval"?

A test yields p = 0.03. Is that the chance the null is true, the chance the result is luck, or something else?

Drug A beats drug B in both mild and severe cases. How many severe patients tip the overall comparison?

Two fitted lines are given with no labels. Only one assignment of roles is possible, and it pins down r.

If two random variables have correlation exactly 0, must they be independent?