Home · Unit 3 · Topic 5

Topic 5: Discrete random variables

General discrete random variables

5 hours

• Understand the concepts of a discrete random variable and its associated probability function, and its use in modelling data. 1 interactive
• Use relative frequencies obtained from data to determine point estimates of probabilities associated with a discrete random variable.
• Recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes. 1 interactive
• Recognise non-uniform discrete random variables and use them to model random phenomena.
• Determine and use the mean (expected value) of a discrete random variable as a measurement of centre, 𝐸(𝑋) = 𝜇 = ∑ 𝑝𝑖 𝑥𝑖 where 𝑝𝑖 is the probability of outcome 𝑥𝑖 occurring. 1 interactive
• Determine and use the variance of a discrete random variable as a measure of spread, 𝑉𝑎𝑟(𝑋) = ∑ 𝑝𝑖 (𝑥𝑖 − 𝜇)2 where 𝑝𝑖 is the probability of outcome 𝑥𝑖 occurring, 𝜇 is the mean. 1 interactive
• Determine and use the standard deviation of a discrete random variable, √𝑉𝑎𝑟(𝑋), as a measure of spread.
• Model and solve problems that involve discrete random variables and associated probabilities, with and without technology.

Bernoulli distributions

2 hours

• Use a Bernoulli random variable as a model for two-outcome situations.
• Identify contexts suitable for modelling by Bernoulli random variables.
• Recognise and determine the mean 𝑝 and variance 𝑝(1 − 𝑝) of the Bernoulli distribution with parameter 𝑝.
• Model and solve problems that involve Bernoulli random variables and associated probabilities, with and without technology.

Binomial distributions

5 hours

• Understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’, 𝑟, in 𝑛 independent Bernoulli trials, with the same probability of success 𝑝 in each trial. 1 interactive
• Identify contexts suitable for modelling by binomial random variables.
• Determine and use the probabilities 𝑃(𝑋 = 𝑟) = (𝑛 𝑟)𝑝𝑟(1 − 𝑝)𝑛−𝑟 associated with the binomial distribution with parameters 𝑛 and 𝑝. 1 interactive
• Calculate the mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) of a binomial distribution using technology and algebraic methods. 1 interactive
• Use the language of probability, including at most, at least, no more than, no less than, inclusive and between.
• Model and solve problems that involve binomial distributions and associated probabilities with and without technology.