5.1 Continuous probability functions

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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 ) = $\frac{1}{20}$ for 0 ≤ x ≤ 20. x = a real number. The graph of f ( x ) = $\frac{1}{20}$ is a horizontal line. However, since 0 ≤ x ≤ 20, f ( x ) is restricted to the portion between x = 0 and x = 20, inclusive.

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.

f ( x ) = $\frac{1}{20}$ for 0 ≤ x ≤ 20.

The graph of f ( x ) = $\frac{1}{20}$ is a horizontal line segment when 0 ≤ x ≤ 20.

The area between f ( x ) = $\frac{1}{20}$ where 0 ≤ x ≤ 20 and the x -axis is the area of a rectangle with base = 20 and height = $\frac{1}{20}$ .

AREA = 20 (\frac{1}{20}) = 1

Suppose we want to find the area between f( x ) = $\frac{1}{20}$ and the x -axis where 0< x <2.

$AREA = (2 - 0) (\frac{1}{20}) = 0.1$

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

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 P (0< x <2) = P ( x <2) = 0.1.

Suppose we want to find the area between f ( x ) = $\frac{1}{20}$ and the x -axis where 4< x <15.

$AREA = (15 - 4) (\frac{1}{20}) = 0.55$

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

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 zero width). Therefore, P ( x = 15) = (base)(height) = (0) $(\frac{1}{20})$ = 0

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 ).

Label the graph with f ( x ) and x . Scale the x and y axes with the maximum x and y values. f ( x ) = $\frac{1}{20}$ , 0 ≤ x ≤ 20.

To calculate the probability that x is between two values, look at the following graph. Shade the region between x = 2.3 and x = 12.7. Then calculate the shaded area of a rectangle.

$P (2.3 < x < 12.7) = (base) (height) = (12.7 - 2.3) (\frac{1}{20}) = 0.52$

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5.1 Continuous probability functions

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