The graphical form of the probability distribution for a continuous random variable x is a smooth curve that might appear as shown in Figure 4.13. This curve, a function of x, is denoted by the symbol f(x) and is variously called a probability density function, a frequency function, or a probability distribution.

The areas under a probability distribution correspond to probabilities for x. For example, the area A beneath the curve between the two points a and b, as shown in Figure 4.13, is the probability that x assumes a value between a and $b(a<x<b).$ Because there is no area over a single point, say, $x=a$, it follows that (according to our model) the probability associated with a particular value of x is equal to 0; that is, $P(x=a)=0,$ and hence $P(a<x<b)=P(a\le x\le b).$ In other words, the probability is the same regardless of whether you include the endpoints of the interval. Also, because areas over intervals represent probabilities, it follows that the total area under a probability distribution—the total probability assigned to the set of all values of x—should equal 1. Note that probability distributions for continuous random variables possess different shapes, depending on the relative frequency distributions of real data that the probability distributions are supposed to model.

Example 4.14 Finding a Continuous Probability—Paper Friction in a Photocopier

Problem

1. Researchers at the University of Rochester studied the friction that occurs in the paper-feeding process of a photocopier and published their results in the Journal of Engineering for Industry. The friction coefficient, x, is a continuous random variable that measures the degree of friction between two adjacent sheets of paper in the feeder stack. The random variable can be modeled using the smooth curve (called a triangular density function) shown in Figure 4.14. Note that the friction coefficient, x, ranges between the values 5 and 15. Find the probability that the friction coefficient is less than 10.

Solution

1. To find the probability that the friction coefficient is less than 10, we need to compute $P(5<x<10)$. This is the area between $x=5$ and $x=10$ shaded on the graph in Figure 4.14. Note that this area is represented by a right triangle with a base of length 5 and a height of .2. Since the area of a triangle is $\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\left(\text{base}\right)\times \left(\text{height}\right)$, then

   [&P|pbo|5|less|x|less|10|pbc||=||pbo|1|sol|2|pbc||pbo|~rom~base|pbc||multi||pbo|height|pbc||=||pbo|~normal~.5|pbc||pbo|5|pbc||pbo|.2|pbc||=|.5 &]

   $$P(5<x<10)=\left(1/2\right)\left(\text{base}\right)\times \left(\text{height}\right)=\left(.5\right)\left(5\right)\left(.2\right)=.5$$

Look Ahead

The areas under most probability distributions are obtained by means of calculus or numerical methods.* Because these methods often involve difficult procedures, we give the areas for 4 common probability distribution in tabular form in Appendix B. Then, to find the area between two values of x, say, $x=a$ and $x=b,$ you simply have to consult the table, or use technology such as statistical software or a graphing calculator.

For the continuous random variable presented in the next section, we will give the formula for the probability distribution, along with its mean $\mu$ and standard deviation $\sigma .$ These two numbers will enable you to make some approximate probability statements about a random variable even when you do not have access to a table of areas under the probability distribution.