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Continuous Random Variables: Continuous Probability Functions

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

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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).

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

We begin by defining a continuous probability density function. We use the function notation fXfX. 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 fXfX 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 fX=120fX=120 for 0≤X≤20 0 X 20. XX = a real number. The graph of fX=120fX=120 is a horizontal line. However, since 0≤X≤20 0 X 20 , fXfX 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.

fX=120fX=120 for 0≤X≤20 0 X 20.

The graph of fX=120fX=120 is a horizontal line segment when 0≤X≤20 0 X 20.

The area between fX=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 fX=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 fX=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. fX=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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