Probability, Mean and Median
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