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 $$\text{SD}(X) = \sqrt{\sigma^2} = \sigma$$