Random variables and sampling

Chapter 60 of 19 strong

RESTS ONState 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 distributionFOUNDATIONSCalculate the meanFOUNDATIONSMultiply and divide fractionsFOUNDATIONSMultiplication and divisionFOUNDATIONSExtract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical formFOUNDATIONSRoundingFOUNDATIONSConstruct direct, contrapositive and contradiction arguments with justified stepsCH 1Substitute known values into a formula and evaluate it correctlyFOUNDATIONSSigned numbersFOUNDATIONSDistinguish a random experiment, one outcome, the full sample space and an event as a subset of that sample spaceFOUNDATIONSSimplify a fractionFOUNDATIONSFor equiprobable outcomes, calculate an event probability as favourable outcomes divided by all outcomes and verify that it lies from zero to oneFOUNDATIONSTrace supplied sequential assignments, decisions and loopsCH 3 Compute a combination coefficient exactly from its factorial ratio, cancelling before multiplication and respecting the selection constraintsCH 4Calculate an exact probability as favourable counts over total counts using the same ordered or unordered model in numerator and denominatorFOUNDATIONSList all possible values of a sample proportion for a stated sample size by listing the possible success counts and dividing each by the sample size, including the boundary values zero and oneFOUNDATIONS

Discrete random-variable summaries

Independent sums and scaling

Simulation design

Sampling distributions

Compute 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 take

Not started yet

Practise this

RESTS ON

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 distributionNot startedCalculate the meanNot startedMultiply and divide fractionsNot started

UNLOCKS

Compute E(X^2) = Σ x^2·Pr(X = x) by squaring each value before multiplying by its probability, and distinguish it from the square of the expected valueCompute Var(X) = E(X^2) − [E(X)]^2 on exact carriers, and interpret the variance as a spread measure in squared unitsProve finite-distribution mean bounds and variance identities with valid probability assumptionsApply E(aX + b) = aE(X) + b and the expectation-of-a-sum rule E(X + Y) = E(X) + E(Y), preserving signs and not scaling the constant by aCalculate finite independent-sum distributions, events and momentsImplement, validate and report a random simulation with justified trial and state rules