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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