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