11–12Mathematics Advanced 11–12 Syllabus (2024)

Record of changes

Implementation from 2026

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Overview Rationale Aim Outcomes Content Assessment Glossary Teaching and learning support

Content

Year 11

Year 12

Teaching advice

Examples

Random variables

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MAV-12-07
solves problems involving discrete probability distributions, continuous random variables and the normal distribution

Random variables

Discrete random variables

Use the relative frequencies of discrete random variable datasets to estimate the probabilities that the random variable $X$ takes each of its values, and explain why these estimated probabilities add to 1
Denote the probability that the discrete random variable $X$ takes the value $x$ by $P (X = x)$ , or by $P (x)$ if the discrete random variable $X$ is understood
Define a discrete probability distribution to be the set of values $x$ taken by a discrete random variable $X$ , together with the probabilities $P (x)$ that $x$ is the outcome of the experiment
Define a discrete random variable to be uniformly distributed if it has Loading many values, all with the same probability, and use it to model random phenomena with Loading
Recognise the mean, or expected value, $μ$ or $E (X)$ , of a discrete random variable $X$ as a measure of centre for its distribution
Generate and use the formulas $E (X) = μ = Σ xP (x)$ and $V a r (X) = σ^{2} = Σ {(x - μ)}^{2} P (x) = Σ x^{2} P (x) - μ^{2}$ for the random variable $X$ , where $Var (X)$ is the variance and $σ$ is the standard deviation of the distribution
Recognise that the variance is an expected value because $Var (X) = E ({(X - μ)}^{2})$
Generate a probability distribution for a given discrete random variable and represent the Loading in graphical and tabular form
Solve problems involving probabilities, expectation and variance of discrete random variables

Continuous random variables

Estimate the probability that a Loading falls in some interval using relative frequencies and histograms obtained from data
Recognise that the probability of a particular value for a continuous random variable is 0 and hence that $P (a < X < b) = P (a \leq X < b) = P (a < X \leq b) = P (a \leq X \leq b)$ since $P (X = a) = P (X = b) = 0$ when $X$ is a continuous random variable
Define the cumulative distribution function (CDF), $F (x)$ , as the probability of a random variable, $X$ , having values less than or equal to $x$ , so $F (x) = P (X \leq x)$ and $P (a \leq x \leq b) = F (b) - F (a)$
Recognise that the cumulative distribution function, $F (x)$ , is non-decreasing for all $x$ in its domain, and graph cumulative distribution functions, given a formula for $F (x)$ , with and without graphing applications
Define a probability density function (PDF), $f (x)$ , for a random variable $X$ with cumulative distribution function $F (x)$ as $f (x) = F' (x)$ and recognise that $P (a < X < b) = \int_{a}^{b} f (x) dx$
Recognise the properties of a probability density function: $f (x) \geq 0$ for all $x$ in the domain of $f (x),$ and $\int_{a}^{b} f (x) dx = 1$ if the domain of $f (x)$ is $[a, b]$ , or $\int_{- \infty}^{\infty} f (x) dx = 1$ if the domain of $f (x)$ is all real $x$
Apply the properties of a probability density function to solve problems and justify conclusions
Find the Loading from a given probability density function
Obtain the cumulative distribution function using the formula $F (x) = \int_{a}^{x} f (t) dt$ where $f (x)$ is a given probability density function defined on the interval $(a, b]$
Determine and use the probability density function $f (x) = \frac{1}{b - a}$ for a continuous uniform distribution for a random variable $X$ taking values in the interval $[a, b]$
Use a cumulative distribution function to calculate the Loading and Loading for a continuous random variable
Find the probability density function from a given cumulative distribution function
Generate the expression for the expected value of a continuous random variable, $E (X) = μ = \int_{a}^{b} xf (x) dx$ , where the probability density function $f (x)$ is defined on the interval $[a, b]$
Generate the expression for the variance of a continuous random variable, $Var (X) = E (X^{2}) - μ^{2} = \int_{a}^{b} x^{2} f (x) dx - μ^{2}$ , where the probability density function $f (x)$ is defined on the interval $[a, b]$
Evaluate the expected value and the variance of a continuous random variable, where the probability density function $f (x)$ is defined on the interval $[a, b]$ , that involve integration of functions within the scope of the Mathematics Advanced course
Evaluate the expected value and the variance of a continuous random variable, where the probability density function $f (x)$ is defined on the interval $[a, b]$ , that involve integration of functions beyond the scope of the Mathematics Advanced course using an online computational application

The normal distribution

Identify the Loading as a continuous probability distribution that is used to model many naturally occurring phenomena
Identify the graph of the probability density function of a normal distribution, the Loading curve, as an ‘ideal’ bell-shaped curve, symmetrical about its mean which is equal to its mode and median, and as having most values concentrated about the mean
Identify contexts that can be approximately modelled by a Loading
Use the notation $X \sim N (μ, σ^{2})$ to represent a normally distributed random variable that has mean $μ$ and standard deviation $σ$
Represent probabilities associated with the normal distribution by areas of shaded regions under the normal curve, which may extend to $\pm \infty$
Apply the Loading to make judgements and solve problems involving probabilities of normally distributed data: that for normal distributions, approximately 68% of data lie within one standard deviation of the mean, approximately 95% within two standard deviations of the mean and approximately 99.7% within three standard deviations of the mean
Use graphing applications to explore the normal distribution, graph the probability density function $f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{({x - μ)}^{2}}{2 σ^{2}}}$ , verify the empirical rule and graph the cumulative distribution function
Recognise features of the normal curve, and identify the Loading and points of inflection
Distinguish between a standard normal distribution with mean 0 and standard deviation 1, and the non-standard normal distribution with mean $μ$ and standard deviation $σ$
Define the $z$ -score, or standardised score, by the formula $z = \frac{x - μ}{σ}$ , where $μ$ is the mean and $σ$ is the standard deviation, and $x$ is an observed value of a random variable
Interpret the $z$ -score as the number of standard deviations a score lies above or below the mean
Use $z$ -scores to compare scores from different sets of data and justify conclusions
Use $z$ -scores to identify probabilities of events less or more extreme than a given outcome and solve problems using tables for the standard normal distribution
Solve problems involving finding the mean or standard deviation of a normal random variable given the probability of an event less or more extreme than a given outcome
Use $z$ -scores to make judgements related to probabilities of certain events or given sets of data assuming an underlying normal distribution

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