distribution_functions - Distribution Functions A...

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D istribution F unctions A distribution function is a function which describes how values are allocated across a population or sample space. There are a variety of settings for which such functions arise and we shall turn our attention to some examples. Example 1: Suppose you flip a fair coin ten times. What is the distribution of the number of heads that you will observe? Plot the distribution. Solution: There are a total of 1024 different combinations of heads and tails that we can observe since we have ten slots to fill and in each slot, there are only two possible values, a Head or a Tail. So, we have 2 ÿ 2 ÿ 2 ÿ 2 ÿ 2 ÿ 2 ÿ 2 ÿ 2 ÿ 2 ÿ 2 = 2 10 = 1024 different outcomes. We could count each of the 1,024 outcomes and record the number of heads. This would take a little while and in the interest of space, the details are omitted. Instead, we end up with the following table: Number of Heads 0 2 3 4 5 6 7 8 9 10 Proportion showing 56 330 462 165 11 that many heads 1024 1024 1024 1024 1024  We can also display our data in a histogram. A histogram is a graphical way to represent data where vertical bars are paced above each category (group) in such a way that the area of each bar represents the proportion of the population in that category. Figure 1: Histogram of Coin Flips 1
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Notice that we grouped the data in only five categories. If we had instead made a table for each coin flip, we could get better estimates. And we might be interested in fitting a smooth curve to the data.
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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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distribution_functions - Distribution Functions A...

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