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Chapter 9: The Normal Probability Distribution
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Next we will discuss how one may view the normal probability distribution
as a limiting case of the binomial distribution. Recall you encountered the
binomial distribution earlier in this e-book.
9 - 2 The Normal Probability Distribution
One can view the normal distribution as the limiting distribution of a
binomial random variable. That is, in a binomial experiment, if we use a
fixed probability of success
, we can analyze what happens as the number
of trials,
, increases. One way to visualize what happens is to construct
histograms for a fixed
and increasingly large
.
Figure 9-2
shows
histograms for
= 5, 10, 20, 50, 100 and 500, with
= 0.12 when the
simulation was done 1,000 times. Superimposed on the histograms, are
smooth curves which show the shapes of the distributions. Observe that as
increases for a fixed
= 0.12, the shape of the smooth curve becomes
increasingly bell-shaped. This bell-shaped curve is associated with the
normal distribution. Similar results can be observed for different
values.
Also observe how the initial right skew pattern of the approximation
becomes more symmetrical as the sample size
increased.