Statistical distributions

Connect an experiment to the values it can produce and their probabilities.

Twelve lessons cover discrete distributions, binomial models, cumulative probabilities, cutoffs and repeated experiments. Work through them in order or choose a topic, then finish with the original mixed practice.

Discrete random variablesDefine a numerical random variable, list its possible values and build a probability distribution by combining the outcomes that produce the same value.
Probability mass functionsCheck probability tables, find normalising constants, handle piecewise formulas and calculate events from a discrete probability mass function.
Discrete uniform distributionsCalculate probabilities on equally likely finite supports, handle integer endpoints and combine an independent uniform variable with a non-uniform one.
Constructing probability distributionsBuild a discrete distribution from an experiment, distinguish a success count from attempts used, and handle the final probability in a capped stopping rule.
Choosing a binomial modelDecide when a count can be modelled by a binomial distribution, define success and identify the fixed-trial, common-probability and independence assumptions.
Binomial probabilitiesDerive the binomial formula by counting success positions, calculate exact-count probabilities and distinguish one ordered path from all qualifying orders.
Cumulative binomial probabilitiesBuild cumulative binomial probabilities by adding point masses, read a cumulative table and choose the correct inclusive cutoff on a calculator.
Binomial tails and intervalsTranslate at least, at most and strict inequalities into binomial events, then use complements and differences of cumulative probabilities without off-by-one errors.
Finding binomial cutoffsFind binomial thresholds that meet a target probability, distinguish smallest and largest cutoffs and prove the boundary using an adjacent integer.
Finding a required number of trialsFind the minimum number of independent trials needed for a target chance of at least one success, using complements, logarithms and adjacent-integer checks.
Repeated binomial experimentsCalculate a within-group binomial probability, use it as the success chance for independent repeated groups and distinguish a binomial count from a fixed offset plus a random count.
Statistical distributions: mixed practicePractise choosing discrete and binomial models, constructing capped distributions, calculating tails, proving cutoffs and combining repeated or shifted counts.