This is Chapter 8 – Continuous Random Variables for the Year 12 Mathematical Methods course.

This chapter extends probability from discrete outcomes to quantities that can take any value within a continuous interval. Instead of assigning probability to individual values, students interpret probability as area under a probability density function, creating an important connection between statistics and integration.

We begin by defining continuous random variables and examining the properties that a valid probability density function must satisfy. Students use definite integrals to calculate probabilities over intervals and must recognise that the probability of one exact value is treated differently from probability across a range.

The normal distribution is then introduced as one of the most important continuous probability models. Students learn how the mean and standard deviation determine its centre and spread and how changes to these parameters affect the overall distribution.

Calculator-based probability work becomes increasingly important. Students determine probabilities, cut-off values and intervals efficiently while maintaining correct notation and explaining what each result means in context. The standard normal distribution and Z-scores provide a common scale for comparing values from different normal distributions. Students must distinguish carefully between entering a lower bound, upper bound or cumulative probability, as calculator errors can produce plausible-looking answers that do not represent the requested region.

Quantiles extend this reasoning by locating values corresponding to stated proportions of a distribution. Continuous probability models are widely used in medicine, quality control, finance, engineering and scientific research, where measurements such as height, time, error and biological variation are naturally treated as continuous.

Subchapters:

• 8A: Continuous random variables
  
• 8B: Probability density functions
• 8C: The normal distribution
• 8D: Probabilities using a calculator
• 8E: The standard normal distribution (Z-distribution)
• 8F: Quantiles

By the end of this chapter, students will be able to analyse continuous probability distributions, use the normal model and Z-scores, and calculate probabilities and quantiles accurately.