The Exam P distributions, in one pass

Learn the model, then the moments

There are about a dozen distributions on the Exam P syllabus and roughly forty facts attached to them. Memorising forty facts is unpleasant and fragile. Learning a dozen models and rebuilding the facts is slower for a week and faster forever, because a model you understand survives exam-room pressure in a way a memorised variance does not.

The test of whether you have learned a family is not whether you can state its variance. It is whether, given a described experiment, you can say which family it is and why - and then, if the formula deserts you, rebuild the mean from the story. A candidate who knows that the geometric distribution counts failures before the first success can reconstruct its mean; a candidate who has only memorised a ratio cannot tell it from the trials version under time pressure.

The discrete families

Five discrete families carry the syllabus, and each answers a different question about the same underlying trials.

Take a binomial with twenty trials and success probability 0.15. The mean is 3, the variance is 2.55, and the standard deviation is about 1.597 - note that the variance is smaller than the mean, which is always true for a binomial and is one of the quickest ways to spot a misidentified family. A Poisson with rate 2.4 has mean and variance both 2.4, and the probability of no events at all is the exponential of minus 2.4, which is 0.0907.

• Discrete uniform: every value in a finite set is equally likely. The model for a fair die or a randomly chosen index.
• Binomial: the number of successes in a fixed number of independent trials with constant success probability.
• Geometric: the number of failures before the first success. Its trials version is this plus one, so the mean shifts by one and the variance does not.
• Negative binomial: the number of failures before the rth success. At r equal to one it is the geometric. With r equal to 3 and success probability 0.25 the mean is 9 and the variance is 36.
• Hypergeometric: successes in a sample drawn without replacement. The binomial's variance times a finite-population correction.
• Poisson: counts in a fixed interval when events arrive independently at a constant rate. Mean and variance are equal, and that equality is diagnostic.

The continuous families

The continuous side splits into the ones built from an exponential and the ones built from a normal, and the split is worth holding in mind because the two halves are manipulated differently.

An exponential with mean 5 gives a probability of exceeding 8 equal to the exponential of minus 1.6, which is 0.2019 - and, by memorylessness, the probability of exceeding 13 given that it already exceeds 5 is the same 0.2019. That property belongs to the exponential alone, and applying it to anything else is a reliable way to lose a mark.

• Continuous uniform: constant density on an interval. Over 0 to 10 the mean is 5 and the variance is 8.333.
• Exponential: waiting time between Poisson events. Memoryless, and the only continuous distribution that is.
• Gamma: the sum of independent exponentials. With shape 4 and rate 0.0004 the mean is 10,000 and the variance is 25,000,000.
• Normal: the limit of standardised sums, and the reason the central limit theorem is on the syllabus at all.
• Lognormal: the exponential of a normal. Its parameters belong to the underlying normal, never to itself.
• Beta: a density on the unit interval, the natural model for a proportion.

The parameterisation traps

Most distribution errors on Exam P are not conceptual. They are parameterisation errors, where the candidate knows the family and uses the wrong version of it, and they are worth rehearsing explicitly because the arithmetic that follows will be flawless and the answer will still be wrong.

A lognormal with underlying parameters 7 and 0.8 has mean 1510.20, which is not the exponential of 7 and is not anything you can read off the parameters without adding half the variance first. A Pareto with shape 3 and scale 2000 has mean 1000 and variance 3,000,000, and its mean does not exist at all if the shape drops to 1.

• Geometric and negative binomial: failures before the success, or trials until it. Read the support the question implies.
• Gamma: shape with a rate, or shape with a scale. The two are reciprocals, and a mean computed with the wrong one is out by the square of the parameter.
• Lognormal: the stated parameters describe the underlying normal, so the mean of the lognormal is the exponential of the mean plus half the variance.
• Normal approximations to discrete distributions: apply the continuity correction, or lose half a unit at every boundary.
• Uniform: some questions give endpoints and some give a mean and a width. Convert once, at the start.

How to drill them

Ten minutes at the start of every study session, answering prompts aloud or on paper before anything else happens. What does a negative binomial count? What is the variance of a Poisson? Which distribution is memoryless? Ten minutes a day for three weeks is about three and a half hours, and it is the highest-yield time in a study plan.

Then, once recall is automatic, switch the drill round: read a described experiment and name the family before computing anything. Identification is the skill the exam actually tests, because the moments follow mechanically once the family is right and no amount of correct arithmetic rescues the wrong family.