Continuous Probability Distributions and the Normal Distribution

Previously, we learned that a random variable is a numerical description of an experiment and that each possible outcome of the experiment is associated with a value of the random variable. In conjunction with the random variable, we can construct the probability distribution of the random variable, which lists all possible values of the random variable along with the probability that the random variable takes on that particular value. In the preceding chapter, we looked at discrete random variables and their associated probability distributions. Because a discrete random variable only takes on certain numerical values, the probability distribution of a discrete random variable is often presented as a table, listing the values of the random variable and their corresponding probabilities.

But what about the probability distribution of a continuous random variable? Recall that a continuous random variable takes on any numerical value in an interval or collection of intervals. Most often, continuous random variables are associated with things that are measured, such as height, weight, temperature, and volume. Continuous random variables have many applications, such as baseball batting averages, IQ scores, the length of time a long-distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores.

As with discrete random variables, a continuous random variable associates a number with the outcome of an experiment, and we are interested in the probability corresponding to the values of the random variable. But unlike a discrete random variable, we cannot make a list of all of the possible values of a continuous random variable. For example, suppose we define a random variable [latex]X[/latex] to be the volume, in litres, of milk in a one-litre milk carton. In this case, the random variable is continuous because the random variable is measuring volume. The random variable [latex]X[/latex] can take on any number between [latex]0[/latex] and [latex]1[/latex]. Because there are an infinite number of numbers between 0 and 1, we simply cannot write them all down. This is true for any continuous random variable—it is impossible to write down all of the possible values associated with the random variable. Consequently, we need to look at and work with the probability distribution for a continuous random variable in a different way than we did with discrete random variables. For a continuous random variable, the probability distribution is most often represented by a graph, and the probabilities associated with the continuous random variable are the corresponding areas under the curve.

“5.1 Introduction to Continuous Random Variables” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.