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Topic 5: Continuous Random Variables and the Normal Distribution

3 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we describe probability for a variable that can take any value in an interval?

A continuous random variable is described by a probability density function, where probability is the area under the curve found by integration.

What a probability density function is, why probability equals area under the curve, the two conditions a valid PDF must satisfy, and how to find probabilities and the mean by integration.

8 min read →

What are the properties of the normal distribution and why is it so widely used?

The normal distribution is a symmetric bell-shaped density defined by its mean and standard deviation, with predictable probabilities given by the empirical rule.

The shape and properties of the normal distribution, the role of the mean and standard deviation, and the 68-95-99.7 empirical rule for estimating probabilities.

8 min read →

How do we find exact normal probabilities for any value using standardisation?

A z-score standardises a normal value to the standard normal distribution, enabling exact probabilities to be found by technology or tables.

How to standardise a normal value with a z-score, interpret z as standard deviations from the mean, and find exact normal probabilities and inverse-normal values.

8 min read →