probability_mean_and_median

probability_mean_and_median - Probability, Mean and Median...

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P robability, M ean and M edian 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 () which is between and ( ) between and b a pxdx xa b p x a b  The cumulative distribution function gives the proportion of the population that has values below t . That is, Proportion of population ( ) having values of below t Pt x 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 ). 2 4 6 8 10 x 0.1 0.2 p 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
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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 is 10 and its height is 0.2. Thus its area is equal to 1 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. 2 4 6 8 10 x 0.1 0.2 p 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 triangle, with a base of 3 and a height of 0.1. Thus, the area is equal to 0.15. A second approach would be to find the equation of the lines that form p ( x ) and use the integral formula on the previous page. For the first line, notice that the line passes through the points (0, 0) and (6, 0.2). Using the point-slope formula, we see that the line is given by p ( x ) = (1/30) x . The second line passes through the points (6, 0.2) and (10, 0). Again, using the point-slope formula, we see that the line is given by p ( x ) = -(1/20) x + 1/2.
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This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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probability_mean_and_median - Probability, Mean and Median...

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