uses concepts and techniques from statistics and probability to present and interpret data and solve problems in a variety of contexts, including the use of probability distributions
uses appropriate technology to investigate, organise, model and interpret information in a range of contexts
provides reasoning to support conclusions which are appropriate to the context
The principal focus of this subtopic is to develop an understanding of discrete random variables and their uses in modelling random processes involving chance.
Students develop understanding and appreciation of probability distributions and associated statistical analysis methods and their use in modelling binomial events.
- define and categorise random variables
- know that a random variable measures some aspect in a population from which samples can be drawn
- know the difference between a discrete random variable and a continuous random variable
- use discrete random variables and associated probabilities to solve practical problems (ACMMM136, ACMMM142)
- use relative frequencies obtained from data to obtain estimates of probabilities associated with a discrete random variable (ACMMM137)
- recognise uniform random variables and use them to model random phenomena having equally likely outcomes (ACMMM138)
- examine simple examples of non-uniform discrete random variables, and recognise that for any random variable, \(X\), the sum of the probabilities is 1 (ACMMM139)
- recognise the mean or expected value, \(E(X)=\mu\), of a discrete random variable \(X\) as a measurement of centre, and evaluate it in simple cases when a discrete probability table is given (by calculating the sum of \(XP(X)\) for all values of \(X\)) (ACMMM140)
- recognise the variance, \(Var(X)\), and standard deviation (\(\sigma\)) of a discrete random variable as measures of spread, and evaluate them in simple cases when a discrete probability table is given, using the formula \(Var(X) = E\left[(X-\mu)^2\right] = E\left(X^2\right) - \mu^2\) (ACMMM141)
- understand that a sample mean, \(\ \bar x\), is an estimate of the associated population mean \(\mu\), and that the sample standard deviation, \(s\), is an estimate of the associated population standard deviation, \(\sigma\), and that these estimates get better as the sample size increases and when we have independent observations