# VCE Mathematical Methods Units 3 and 4 – Probability and Statistics

4.1 Descriptive Statistics
4.2 Discrete Random Variables
4.3 Binomial Distribution
4.4 Statistical Inference

In this area of study students cover discrete and continuous random variables, their representation using tables, probability functions (specified by rule and defining parameters as appropriate); the calculation and interpretation of central measures and measures of spread; and statistical inference for sample proportions. The focus is on understanding the notion of a random variable, related parameters, properties and application and interpretation in context for a given probability distribution.

This area of study includes:

• random variables, including the concept of a random variable as a real function defined on a sample space and examples of discrete and continuous random variables
• discrete random variables:
• specification of probability distributions for discrete random variables using graphs, tables and probability mass functions
• calculation and interpretation and use of mean $(\mu)$, variance $(\sigma^2)$ and standard deviation of a discrete random variable and their use
• bernoulli trials and the binomial distribution, $\text{Bi}(n, p)$, as an example of a probability distribution for a discrete random variable
• effect of variation in the value/s of defining parameters on the graph of a given probability mass function for a discrete random variable
• calculation of probabilities for specific values of a random variable and intervals defined in terms of a random variable, including conditional probability
continuous random variables:
• construction of probability density functions from non-negative functions of a real variable
• specification of probability distributions for continuous random variables using probability density functions
• calculation and interpretation of mean $(\mu)$, median, variance $(\sigma^2)$ and standard deviation of a continuous random variable and their use
• standard normal distribution, $\text{N}(0, 1)$, and transformed normal distributions, $\text{N}(\mu, \sigma^2 )$, as examples of a probability distribution for a continuous random variable
• effect of variation in the value/s of defining parameters on the graph of a given probability density function for a continuous random variable
• calculation of probabilities for intervals defined in terms of a random variable, including conditional probability (the cumulative distribution function may be used but is not required)
• Statistical inference, including definition and distribution of sample proportions, simulations and confidence intervals:
• distinction between a population parameter and a sample statistic and the use of the sample statistic to estimate the population parameter
• concept of the sample proportion $\displaystyle \hat{P} = \frac{X}{n}$ as a random variable whose value varies between samples, where $X$ is a binomial random variable which is associated with the number of items that have a particular characteristic and $n$ is the sample size
• approximate normality of the distribution of $\hat{P}$ for large samples and, for such a situation, the mean $p$, (the population proportion) and standard deviation, $\displaystyle \sqrt{\frac{p(1-p)}{n}}$
• simulation of random sampling, for a variety of values of $p$ and a range of sample sizes, to illustrate the distribution of $\hat{P}$
• determination of, from a large sample, an approximate confidence interval $\displaystyle \left( \hat{p}-z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\hat{p}+z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right)$ for a population proportion where z is the appropriate quantile for the standard normal distribution, in particular the 95% confidence interval as an example of such an interval where $z \approx 1.96$ (the term standard error may be used but is not required).

source – VCE Mathematics Study Design

VCE Mathematical Methods Units 3 and 4 Courses

VCE Mathematical Methods Units 3 and 4 Syllabus

## Course Content

### Descriptive Statistics

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Discrete Random Variables
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Continuous Random Variables
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Binomial Distribution
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• 18 Lessons
• 75 Topics
• 24 Quizzes