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In the last section, we considered (probability) density functions. We went on to discuss their relationship with cumulative distribution functions. The goal of this section is to take a closer look at densities, introduce some common distributions and discuss the mean and median. Recall, we define probabilities as follows:

Proportion of population for Area under the graph of p ( x ) between a and b which x is between a and b

p( x)dx a b

The cumulative distribution function gives the proportion of the population that has values below t. That is, t P (t )

p( x)dx

Proportion of population having values of x below t

When answering some questions involving probabilities, both the density function and the cumulative distribution can be used, as the next example illustrates.

Example 1:

Consider the graph of the function p(x). p x

0.2 0.1

2

4

6

8

10

x

Figure 1: The graph of the function p(x)

a. Explain why the function is a probability density function. b. Use the graph to find P(X < 3) c. Use the graph to find P(3 § X § 8)

1

Solution:

a. Recall, a function is a probability density function if the area under the curve is equal to 1 and all of the values of p(x) are non-negative. It is immediately clear that the values of p(x) are non-negative. To verify that the area under the curve is equal to 1, we recognize that the graph above can be viewed as a triangle. Its base 1 is 10 and its height is 0.2. Thus its area is equal to 10 0.2 1 . 2

b. There are two ways that we can solve this problem. Before we get started, though, we begin by drawing the shaded region. p x

0.2 0.1

2

4

6

8

10

x

The first approach is to recognize that we can determine the area under the curve from 0 to 3 immediately. The shaded area is another…...

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