## Assumed knowledge

• The content of the modules:
• A basic understanding of integration, as discussed in the module Integration .

## Motivation

The module Discrete probability distributions introduces the fundamentals of random variables, noting that they are the numerical outcome of a random procedure.

Most of the examples considered in that module involve counts of some sort: the number of things, or people, or occurrences, and so on. When we count, the outcome is definitely discrete: it can only take integer values, and not other numerical values.

However, many measured variables are not like this. Rather, they take a value in a specified range (for example, a variable might be positive), but within that range they can take any numerical value. The paradigm phenomenon in this category is 'time'. If we ask the question 'How long does it take?' (to complete a crossword, to brush your teeth, for a person to die after a diagnosis of an illness, to run 10 kilometres), then the answer can be given to varying levels of accuracy, and we are really only limited by the precision of our instruments. We feel that, in principle, the time could be any numerical value. For example, the time take to run 10 kilometres could be 50 minutes and 23.1 seconds. But it could also be 50 minutes and 23.08 seconds, or 50 minutes and 23.082 seconds, and so on.

Many other variables are measured on a continuum like this. These variables include height, weight, blood pressure, temperature, distance, speed and many others. We need a way to represent the probability distribution of such continuous variables, and the purpose of this module is to describe this.

There are different ways to describe the probability distribution of a continuous random variable. In this module, we introduce the cumulative distribution function and the probability density function. We shall see that probabilities associated with a continuous random variable are given by integrals. This module also covers the mean and variance of a continuous random variable.

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