5.1: Continuous Probability Functions

Learning Outcomes

  • Recognize and understand continuous probability density functions in general

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

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

[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

[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

[latex]P(X{\leq}x)[/latex] (can be written as [latex]P(Xx)[/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

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]

Please watch this video to 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

Concept Review

The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points
a and b is equal to [latex]P(aX is a continuous random variable, the probability density function (pdf), f(x), is used to draw the graph of the probability distribution. The total area under the graph of f(x) is one. The area under the graph of f(x) and between values a and b gives the probability [latex]P(a

The cumulative distribution function (cdf) of X is defined by P (X ≤ x). It is a function of x that gives the probability that the random variable is less than or equal to x.

Formula Review

Probability density function (pdf) f(x):

  • [latex]f(x){\geq}0[/latex]
  • The total area under the curve f(x) is one.

Cumulative distribution function (cdf): [latex]P(X{\leq}x)[/latex]