What is a Continuous Probability Function?

Learning Outcomes

  • Draw a continuous probability function for a uniform distribution
  • Calculate a probability for a uniform distribution

Recall: Inequality Symbols

An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than.

Here are some common inequalities seen in statistics:

  • < indicates less than, for example x < 5 indicates x is less than 5
  • ≤ indicates less than or equal to, for example x ≤ 5 indicates x is less than or equal to 5 (5 is included)
  • > indicates greater than, for example x > 5 indicates x is greater than 5
  • ≥ indicates greater than or equal to, for example x ≥ 5 indicates x is greater than or equal to 5 (5 is included)

Note: Where you place the variable in the inequality statement can change the symbol you use.

For example:

  • x < 5 indicates all possible numbers less than 5.
  • 5 < x indicates that 5 is less than x, or we could rewrite this with the x on the left: x > 5.

Note: how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.

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.

Recall: Area of a Rectangle

Area is a measure of the surface covered by a figure. A rectangle has four sides, the figure below is an example where [latex]W[/latex] is the width and [latex]L[/latex] is the length.

A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled [latex]L[/latex], the sides are labeled [latex]W[/latex].

The area, [latex]A[/latex], of a rectangle is the length times the width.

[latex]A=L \cdot W[/latex]

Example

Consider the function [latex]f(x)\displaystyle\frac{{1}}{{20}}[/latex] is a horizontal line. However, since [latex]0{\leq}x{\leq}20[/latex], f(x) is restricted to the portion between [latex]x=0[/latex] and [latex]x=20[/latex], inclusive.

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). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle.

[latex]f(x)=\frac{{1}}{{20}}[/latex] for [latex]0{\leq}x{\leq}20[/latex].

The graph of [latex]f(x)=\frac{{1}}{{20}}[/latex] is a horizontal line segment when [latex]0{\leq}x{\leq}20[/latex].

The area between [latex]f(x)\frac{{1}}{{20}}[/latex].

[latex]\displaystyle\text{AREA}={20}{(\frac{{1}}{{20}})}={1}[/latex]

Suppose we want to find the area between [latex]f(x)=[/latex] and the x-axis where [latex]0<x<2[/latex].

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). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 0 to x = 2.

[latex]\displaystyle\text{AREA}={({2}-{0})}{(\frac{{1}}{{20}})}={0.1}[/latex]

[latex]\displaystyle({2}-{0})={2}=\text{base of a rectangle}[/latex]

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 [latex]P(0<x<2)=P(x<2)=0.1[/latex].

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

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). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 4 to x = 15.

[latex]\displaystyle\text{AREA}={({15}-{4})}{(\frac{{1}}{{20}})}={0.55}[/latex]

[latex]\displaystyle\text{AREA}={({15}-{4})}{(\frac{{1}}{{20}})}={0.55}[/latex]

[latex]\displaystyle{({15}-{4})}={11}=\text{the base of a rectangle}[/latex]

The area corresponds to the probability [latex]P(4<x<15)=0.55[/latex].

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

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). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A vertical line extends from the horizontal axis to the graph at x = 15.

[latex]P(X{\leq}x)[/latex] (can be written as [latex]P(X<x)[/latex] for continuous distributions) is called the cumulative distribution function or CDF. Notice the “less than or equal to” symbol. We can use the CDF to calculate [latex]P(X>x)[/latex]. The CDF gives “area to the left” and [latex]P(X>x)[/latex] gives “area to the right.” We calculate [latex]P(X > x)[/latex] for continuous distributions as follows: [latex]P(X>x)=1–P(X<x)[/latex].

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). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. The area to the left of a value, x, is shaded.

Label the graph with f(x) and x. Scale the x and y axes with the maximum x and y values.f(x) = [latex]\displaystyle\frac{{1}}{{20}}[/latex], [latex]0{\leq}x{\leq}20[/latex].

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

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). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 2.3 to x = 12.7

[latex]\displaystyle{P}{({2.3}{<}{x}{<}{12.7})}={(\text{base})}{(\text{height})}={({12.7}-{2.3})}{(\frac{{1}}{{20}})}={0.52}[/latex]

This video will help you summarize what you just read.

Try It

Consider the function [latex]f(x)\frac{{1}}{{8}}[/latex] for [latex]0{\leq}x{\leq}8[/latex]. Draw the graph of f(x) and find [latex]P(2.5<x<7.5)[/latex].