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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 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.

Consider the function f x = 1 20 for 0 x 20 . x = a real number. The graph of f x = 1 20 is a horizontal line. However, since 0 x 20 , f x is restricted to the portion between x 0 and x 20 , 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 = 1 20 for 0 x 20 .

The graph of f x = 1 20 isa horizontal line segment when 0 x 20 .

The area between f x = 1 20 where 0 x 20 and the x-axis is the area of a rectangle with base = 20 and height = 1 20 .

AREA 20 1 20 1

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

Suppose we want to find the area between f x = 1 20 and the x-axis where 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 ) 1 20 0.1

( 2 - 0 ) 2 base of a rectangle

1 20 = the height.

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

Suppose we want to find the area between f x = 1 20 and the x-axis where 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 ) 1 20 0.55

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

1 20 = the height.

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

Suppose we want to find P ( x = 15 ) . On an x-y graph, x = 15 is a vertical line. A vertical line has no width (or 0 width). Therefore, P ( x = 15 ) = (base) (height) = ( 0 ) ( 1 20 ) = 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 ) (can be written as P ( X x ) 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 P ( X x ) . The CDF gives "area to the left" and P ( X x ) gives "area to the right." We calculate P ( X x ) for continuous distributions as follows: 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) and x . Scale the x and y axes with the maximum x and y values. f x 1 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 ) ( 1 20 ) 0.52

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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