We have just discussed various ways of assigning probability values to the outcomes of an experiment. Let us now use this probability information to compute the expected outcome, variance, and standard deviation of the experiment. This can help select the best decision among a number of alternatives.

A random variable assigns a real number to every possible outcome or event in an experiment. It is normally represented by a letter such as X or Y. When the outcome itself is numerical or quantitative, the outcome numbers can be the random variable. For example, consider refrigerator sales at an appliance store. The number of refrigerators sold during a given day can be the random variable. Using X to represent this random variable, we can express this relationship as follows:

In general, whenever the experiment has quantifiable outcomes, it is beneficial to define these quantitative outcomes as the random variable. Examples are given in Table 2.5.

When the outcome itself is not numerical or quantitative, it is necessary to define a random variable that associates each outcome with a unique real number. Several examples are given in Table 2.6.

There are two types of random variables: discrete random variables and continuous random variables. Developing probability distributions and making computations based on these distributions depend on the type of random variable.

A random variable is a discrete random variable if it can assume only a finite or limited set of values. Which of the random variables in Table 2.5 are discrete random variables? Looking at Table 2.5, we can see that the variables associated with stocking 50 Christmas trees, inspecting 600 items, and sending out 5,000 letters are all examples of discrete random variables. Each of these random variables can assume only a finite or limited set of values. The number of Christmas trees sold, for example, can only be integer numbers from 0 to 50. There are 51 values that the random variable X can assume in this example.

A continuous random variable is a random variable that has an infinite or an unlimited set of values. Are there any examples of continuous random variables in Table 2.5 or 2.6? Looking at Table 2.5, we can see that testing the lifetime of a lightbulb is an experiment whose results can be described with a continuous random variable. In this case, the random variable, S, is the time the bulb burns. It can last for 3,206 minutes, 6,500.7 minutes, 251.726 minutes, or any other value between 0 and 80,000 minutes. In most cases, the range of a continuous random variable is stated as lower value$\le S\le$upper value, such as $0\le S\le 80,000.$ The random variable R in Table 2.5 is also continuous. Can you explain why?