Continuous Probability Functions and the Uniform Distribution

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

Figure 5.5

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

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

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

$AREA = 20(\dfrac{1}{20})=1$

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

Figure 5.6

$AREA = (2 – 0)(\dfrac{1}{20}) = 0.1$

$(2–0)=2=\text{base of a rectangle}$

Reminder

$\text{area of a rectangle = (base)(height)}$

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 $f(x) = \dfrac{1}{20}$ and the x-axis where 4 < x < 15.

Figure 5.7

$\text{AREA = }(15 – 4)(\dfrac{1}{20}) = 0.55$

(15 – 4) = 11 = the base of a rectangle

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, $\text{ P(x = 15) = (base)(height) }= (0)(\dfrac{1}{20}) = 0$

Figure 5.8

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).

Figure 5.9

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

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.

Figure 5.10

$\text{ P (2.3 < x < 12.7) = (base)(height) }=(12.7−2.3)(\dfrac{1}{20})=0.52$

Try It 5.1

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

Source: OpenStax, https://openstax.org/books/statistics/pages/5-introduction
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