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# Continuous Probability Functions

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 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 0x20 0 x 20. xx = a real number. The graph of fx=120fx=120 is a horizontal line. However, since 0x20 0 x 20 , fxfx is restricted to the portion between x=0x0 and x=20x20, inclusive .

fx=120fx=120 for 0x20 0 x 20.

The graph of fx=120fx=120 is a horizontal line segment when 0x20 0 x 20.

The area between fx=120fx=120 where 0x20 0 x 20 and the x-axis is the area of a rectangle with base = 2020 and height =120120.

AREA=20120=1 AREA 20120 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 .

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 .

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.

P(Xx) 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).

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, 0x20 0 x 20.

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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