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

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND