When a random variable may assume any numerical value in one or more intervals on the real number line, then the random variable is called a **continuous random variable**. For example, the EPAcombined city and highway mileage of a randomly selected midsize car is a continuous random variable. Furthermore, the temperature (in degrees Fahrenheit) of a randomly selected cup of coffee at a fast-food restaurant is also a continuous random variable. We often wish to compute probabilities about the range of values that a continuous random variable x might attain. For example, suppose that marketing research done by a fast-food restaurant indicates that coffee tastes best if its temperature is between 153° F and 167° F. The restaurant might then wish to find the probability that x, the temperature of a randomly selected cup of coffee at the restaurant, will be between 153° and 167°. This probability would represent the proportion of coffee served by the restaurant that has a temperature between 153° and 167°. Moreover, one minus this probability would represent the proportion of coffee served by the restaurant that has a temperature outside the range 153° to 167°.

In general, to compute probabilities concerning a continuous random variable x, we assign probabilities to intervals of values by using what we call a **continuous probability distribution**. To understand this idea, suppose that f (x) is a continuous function of the numbers on the real line, and consider the continuous curve that results when f (x) is graphed. Such a curve is illustrated in Figure 6.1. Then:

**Continuous Probability Distributions**

The curve f(x) is the continuous probability distribution of the random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval. Sometimes we refer to a continuous probability distribution as a probability curve or as a probability density function.

An *area* under a continuous probability distribution (or probability curve) is a probability. For instance, consider the range of values on the number line from the number a to the number b—that is, the interval of numbers from a to b. If the continuous random variable x is described by the probability curve f(x), then the area under f(x) corresponding to the interval from a to b is the probability that x will attain a value between a and b. Such a probability is illustrated as the shaded area in Figure 6.1. We write this probability as P(a ≤x≤ b). For example, suppose that the continuous probability curve f(x) in Figure 6.1 describes the random variable x = the temperature of a randomly selected cup of coffee at the fast-food restaurant. It then follows that P(153≤ x ≤167)- the probability that the temperature of a randomly selected cup of coffee at the fast-food restaurant will be between 153° and 167°—is the area under the curve f(x) between 153 and 167.

We know that any probability is 0 or positive, and we also know that the probability assigned to all possible values of x must be 1. It follows that, similar to the conditions required for a discrete probability distribution, a probability curve must satisfy the following properties:

**Properties of a Continuous Probability Distribution**

The **continuous probability distribution** (or **probability curve**) f(x) of a random variable x must satisfy the following two conditions: 1 f (x)≥0 for any value of x. 2 The total area under the curve f (x) is equal to 1. Any continuous curve f (x) that satisfies these conditions is a valid continuous probability distribution. Such probability curves can have a variety of shapes—bell-shaped and symmetrical, skewed to the right, skewed to the left, or any other shape. In a practical problem, the shape of a probability curve would be estimated by looking at a frequency (or relative frequency) histogram of observed data. Later in this article, we study probability curves having several different shapes.

We have seen that to calculate a probability concerning a continuous random variable, we must compute an appropriate area under the curve f (x). In theory, such areas are calculated by calculus methods and/or numerical techniques. Because these methods are difficult, needed areas under commonly used probability curves have been compiled in statistical tables. As we need them, we show how to use the required statistical tables. Also, note that since there is no area under a continuous curve at a single point, the probability that a continuous random variable x will attain a single value is always equal to 0. It follows that in Figure 6.1 we have P(x=a) = 0 and P(x = b)=0. Therefore, P(a ≤ x ≤ b) equals P(a < x < b) because each of the interval endpoints a and b has a probability that is equal to 0.