We begin by defining a continuous probability density function. We use the function notation *f*(*x*). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function *f*(*x*) so that the area between it and the *x*-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. **For continuous probability distributions, PROBABILITY = AREA.**

### Example 5.1

Consider the function
*f*(*x*) = $\frac{1}{20}$ for 0 ≤ *x* ≤ 20. *x* = a real number. The graph of
*f*(*x*) = $\frac{1}{20}$ is a horizontal line. However, since 0 ≤ *x* ≤ 20, *f*(*x*) is restricted to the portion between *x* = 0 and *x* = 20, inclusive.

*f*(*x*) = $\frac{1}{20}$ **for** 0 ≤ *x* ≤ 20.

The graph of *f*(*x*) = $\frac{1}{20}$ is a horizontal line segment when 0 ≤ *x* ≤ 20.

The area between *f*(*x*) = $\frac{1}{20}$ where 0 ≤ *x* ≤ 20 and the *x*-axis is the area of a rectangle with base = 20 and height = $\frac{1}{20}$.

**Suppose we want to find the area between f(x) = $\frac{1}{20}$ and the x-axis where 0 < x < 2.**

$$\text{AREA}=\text{}(2\text{}\u2013\text{}0)\left(\frac{1}{20}\right)\text{}=\text{}0.1$$

$$(2\text{}\u2013\text{}0)\text{}=\text{}2\text{}=\text{base of a rectangle}$$

### Reminder

area of a rectangle = (base)(height)

The area corresponds to a probability. The probability that *x* is between zero and two is 0.1, which can be written mathematically as *P*(0 < *x* < 2) = *P*(*x* < 2) = 0.1.

**Suppose we want to find the area between f(x) = $\frac{1}{20}$ and the x-axis where 4 < x < 15.**

\text{AREA}=\text{}(15\text{}\u2013\text{}4)\left(\frac{1}{20}\right)\text{}=\text{}0.55

$(15\text{}\u2013\text{}4)\text{}=\text{}11\text{}=\text{thebaseofarectangle}$

The area corresponds to the probability *P*(4 < *x* < 15) = 0.55.

Suppose we want to find *P*(*x* = 15). On an *x*-*y* graph, *x* = 15 is a vertical line. A vertical line has no width (or zero width). Therefore, *P*(*x* = 15) = (base)(height) = (0)$\left(\frac{1}{20}\right)$ = 0

*P*(*X* <= *x*), which can also be written as *P*(*X* < *x*) for continuous distributions, is called the cumulative distribution function or CDF. Notice the *less than or equal to* symbol. We can also use the CDF to calculate *P*(*X* > *x*). The CDF gives *area to the left* and *P*(*X* > *x*) gives *area to the right*. We calculate *P*(*X* > *x*) for continuous distributions as follows: *P*(*X* > *x*) = 1 – *P* (*X* < *x*).

Label the graph with
*f*(*x*) and *x*. Scale the *x* and *y* axes with the maximum *x* and *y* values. *f*(*x*) = $\frac{1}{20}$, 0 ≤ *x* ≤ 20.

To calculate the probability that *x* is between two values, look at the following graph. Shade the region between *x* = 2.3 and *x* = 12.7. Then calculate the shaded area of a rectangle.

$P(2.3<x<12.7)=(\text{base})(\text{height})=(12.7-2.3)\left(\frac{1}{20}\right)=0.52$

Consider the function *f*(*x*) = $\frac{\text{1}}{8}$
for 0 ≤ *x* ≤ 8. Draw the graph of *f*(*x*) and find *P*(2.5 < *x* < 7.5).