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Continuous Probability Functions

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module introduces the continuous probability function and explores the relationship between the probability of X and the area under the curve of f(X).

We begin by defining a continuous probability density function. We use the function notation f⁢xfx. 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⁢xfx 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 1

Consider the function f⁢x=120fx=120 for 0≤x≤20 0 x 20. xx = a real number. The graph of f⁢x=120fx=120 is a horizontal line. However, since 0≤x≤20 0 x 20 , f⁢xfx is restricted to the portion between x=0x0 and x=20x20, inclusive .

f(x)=1/20 graph displaying a boxed region consisting of a horizontal line extending to the right from point 1/20 on the y-axis, a vertical upward line from point 20 on the x-axis, and the x and y-axes.

f⁢x=120fx=120 for 0≤x≤20 0 x 20.

The graph of f⁢x=120fx=120 is a horizontal line segment when 0≤x≤20 0 x 20.

The area between f⁢x=120fx=120 where 0≤x≤20 0 x 20 and the x-axis is the area of a rectangle with base = 2020 and height =120120.

AREA=20⋅120=1 AREA 20⋅120 1

This particular function, where we have restricted xx so that the area between the function and the x-axis is 1, is an example of a continuous probability density function. It is used as a tool to calculate probabilities.

Suppose we want to find the area between f⁢x=120fx=120 and the x-axis where 0<x<2 0 x 2 .

f(X)=1/20 graph displaying a boxed region consisting of a horizontal line extending to the right from point 1/20 on the y-axis, a vertical upward line from point 20 on the x-axis, and the x and y-axes. A shaded region ranging from points 0-2 on the x-axis occurs within this area.

AREA=(2-0)⋅ 120=0.1 AREA (2-0)⋅ 120 0.1

(2-0)=2= base of a rectangle (2-0) 2 base of a rectangle

120120 = the height.

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

Suppose we want to find the area between f⁢x=120fx=120 and the x-axis where 4<x<15 4 x 15 .

f(X)=1/20 graph displaying a boxed region consisting of a horizontal line extending to the right from point 1/20 on the y-axis, a vertical upward line from point 20 on the x-axis, and the x and y-axes. A shaded region ranging from points 4-15 on the x-axis occurs within this area.

AREA=(15-4)⋅ 120=0.55 AREA (15-4)⋅ 120 0.55

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

120120 = the height.

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

Suppose we want to find P(x = 15)P(x = 15). On an x-y graph, x = 15x = 15 is a vertical line. A vertical line has no width (or 0 width). Therefore, P(x = 15) = (base)(height) = (0)(120) = 0P(x = 15)=(base)(height)= (0)(120)=0.

f(X)=1/20 graph displaying a boxed region consisting of a horizontal line extending to the right from point 1/20 on the y-axis, a vertical upward line from point 20 on the x-axis, and the x and y-axes. A vertical upward line is drawn from point 15 on the x-axis to the horizontal line occurring from point 1/20 on the y-axis.

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

f(X) graph displaying a boxed region consisting of a horizontal line extending to the right from midway on the y-axis, a vertical upward line from an arbitrary point on the x-axis, and the x and y-axes. A shaded region from points 0-x occurs within this area.

Label the graph with f(x)f(x) and xx. Scale the x and y axes with the maximum xx and yy values. f⁢x=120 fx 120, 0≤x≤20 0 x 20.

f(X) graph displaying a boxed region consisting of a horizontal line extending to the right from midway on the y-axis, a vertical upward line from an arbitrary point on the x-axis, and the x and y-axes. A shaded region from points 2.3-12.7 occurs within this area.

P(2.3<x<12.7)=(base) (height)=(12.7-2.3) (120)=0.52 P(2.3 x 12.7) (base) (height) (12.7-2.3) (120) 0.52

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