Discrete random variables

Chapter 100 of 32 strong

RESTS ONFor equiprobable outcomes, calculate an event probability as favourable outcomes divided by all outcomes and verify that it lies from zero to oneCH 12 Solve a one-step linear equationCH 7 Identify a, b, c in zero form and calculate b squared minus 4ac with signs intactCH 1Construct or recognise a complete finite sample space, preserving order where successive stages make ordered tuples distinctFOUNDATIONSHighest common factorFOUNDATIONSInterpret unions, intersections and complements as “at least one”, “both”, “only”, “neither” and related event descriptions relative to a stated sample spaceFOUNDATIONSComplete and read a two-event count or probability table, including four interior regions, row and column margins and the grand totalFOUNDATIONSDecimal place valueFOUNDATIONS

Discrete random variable properties

Distributions from finite experiments

Expected value, variance and standard deviation

Discrete distribution events and transformations

State and apply the two defining properties of a discrete probability distribution — every probability is in the interval from zero to one inclusive and the probabilities over all values sum exactly to one — using the properties to validate or reject a distribution

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Practise this

RESTS ON

Map each elementary outcome to a stated finite random-variable value, combine the masses of outcomes with the same value, and complete or read its probability tableNot startedComplete and validate a finite probability assignment: every probability is non-negative and the total is exactly one; use stated unequal outcome masses rather than assuming equiprobabilityNot started

UNLOCKS

Evaluate a probability mass function rule at each stated value of its discrete domain, construct the corresponding distribution table, and identify the discrete domain of allowed values rather than treating the rule as a continuous curveSolve for an unknown parameter, or several parameters, in a discrete distribution from the total-probability-one condition and any stated mean, then reject solutions that make any probability invalidCompute the expected value or mean E(X) = Σ x·Pr(X = x) of a discrete random variable from a table or probability mass function, and interpret it as a long-run average rather than necessarily a value the variable can takeCalculate the probability of an interval or inequality event on a discrete random variable by summing exactly the probabilities of the values that satisfy the event, including strict/non-strict boundary treatment and complement useConstruct the two-point Bernoulli distribution table from the success probability, and compute E(X) = p, Var(X) = p(1 − p) and SD(X) = √(p(1 − p)) CH 11Construct the full binomial distribution table for every success count from zero to the trial count using Pr(X = x) = C(n, x)(1 − p)^(n − x)p^x, and verify the exact total is one CH 11Distinguish a continuous random variable from a discrete one, interpret a probability density function as a smooth limiting model of grouped frequency data, and state that Pr(X = x) = 0 at a single value so strict or non-strict interval endpoints do not change an interval probability CH 12Convert normal classification intervals into a discrete probability distribution over three or more stated categories, complete the category probability table so the probabilities sum exactly to one, and compute an expected value, mean profit or mean cost from the completed table CH 13