Distribution Functions A distribution functionis 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 2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2 = 210= 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 Heads02345678910Proportion showing5633046216511that many heads10241024102410241024We can also display our data in a histogram. A histogramis 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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