Continuous Probability Functions and the Uniform Distribution

Once you have grasped the notion of a discrete random variable, it should be clear that all random variables need not be discrete. For example, consider measuring the atmospheric temperature at some prescribed location. The measured temperature would be random and could take on a continuum of values (theoretically speaking). Under these circumstances, we say that the random variable is continuous. All the machinery developed for discrete random values (such as expected value, variance, and mean) must be elevated to continuous random variables to handle this situation. The uniform distribution (which you have programmed using the random module) is an example of a continuous probability distribution.

Continuous Probability Functions

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.

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

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.

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

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

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

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).

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

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.