3.1 Continuous random variables

Recall from Topic 1 that a continuous variable is one where "the variable can take an infinite number of values within a certain range. For example, height, weight or age." Consequentially, height, weight, and age are all examples of variables that are continuous random variables.

Consider the following: let $X$ denote the height in cm of university students. Considering the fact that a height cannot be negative, we could say that the possible values for $X$ are in the range $(0, \infty)$ . ( $\infty$ is a mathematical symbol that means 'infinity'.) A height could be any number within that range. It would be impossible to write down all of the potential values $X$ could take, because $X$ is a continuous random variable and can therefore take an infinite number of values within this range.

Recall that in Topic 2 we discussed the concepts of mean, variance, and standard deviation. These concepts can be related to a continuous random variable $X$ as follows:

Expected value, variance and standard deviation of a random variable

The expected value (or mean) of $X$ can be denoted $\text{E}(X) = \mu$
The variance of $X$ can be denoted $\text{Var}(X) = \sigma^2$
The standard deviation of $X$ is the square root of the variance and can be denoted $\text{SD}(X) = \sqrt{\sigma^2} = \sigma$