Continuous probability distributions
Chapter 120 of 39 strong
RESTS ONMap 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 tableCH 10Complete 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 equiprobabilityCH 10State the complete anti-derivative family with one arbitrary additive constant, and verify that the constant disappears under differentiationCH 7Substitute known values into a formula and evaluate it correctlyCH 4Solve two-step linear equations by inverse operations in reverse orderCH 1Choose class intervals for grouped dataFOUNDATIONSCalculate percentage frequencyFOUNDATIONSConstruct or read a bar chartFOUNDATIONSConstruct or recognise a complete finite sample space, preserving order where successive stages make ordered tuples distinctFOUNDATIONSInterpret the event after the conditioning event is known and restrict the denominator to that event's outcomes or table marginCH 10Determine a unique constant by substituting an exact point or initial value into a general anti-derivative, then evaluate the resulting function when requiredCH 7Connect f'(x), dy/dx, a derivative with respect to a named independent variable, and evaluation notation at a stated input, while treating each as notation for a function or value rather than ordinary fraction arithmeticCH 6Anti-differentiate a x^n by increasing the exponent by one and dividing the exact coefficient by that new exponent, with the stated exclusion n = -1CH 7Anti-differentiate an exact constant multiple, sum or difference term by term while preserving every sign and rational coefficientCH 7Identify the inner and outer functions of a composite, differentiate each, multiply the outer derivative evaluated at the inner function by the inner derivative, and write the result in the requested factorised formCH 6Simplify products, quotients and powers with a common positive base by adding, subtracting or multiplying exponents as appropriate, without applying an index law across addition or subtractionCH 7Differentiate e^x, e^(kx) and e^(g(x)) with the factor of the inner derivative, and differentiate sums, expanded squares and quotient forms after rewriting them as sums of e^(kx) termsCH 5Recall and apply the exact derivative of sine, cosine and tangent components, including constant multiples and linear input arguments, as a component in a larger differentiation taskCH 6Obtain exact sine, cosine and tangent values for the standard first-quadrant angles, their symmetric points and axis boundariesCH 1Form or recognise the union, intersection, difference and complement of finite or interval-defined sets relative to an explicitly stated universeFOUNDATIONSEvaluate a piecewise linear rule in the branch containing the inputFOUNDATIONSRead a piecewise linear graph for the applicable rule, endpoints and valuesFOUNDATIONSCompute Var(X) = E(X^2) − [E(X)]^2 on exact carriers, and interpret the variance as a spread measure in squared unitsCH 10Extract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical formCH 2Completely factorise a low-degree polynomial by taking out common factors, using familiar quadratic patterns, or finding one linear factor and dividing to obtain a lower-degree factorCH 1Solve equations with unknowns on both sides by collecting them onto one sideCH 1Identify a, b, c in zero form and calculate b squared minus 4ac with signs intactCH 1Compute 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 takeCH 10
Continuous random variables and density functions
Probability density validity and normalisation
Interval and conditional probabilities
Mean, variance, standard deviation and percentiles
Linear transformations and independent sums