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Suppose we are interested in counting
the number of times a Bernoulli event with
probability happens in a fixed number of independent trials.
For example, we might be interested in counting
the total number of times hurricanes hit Florida
out of a specified number of hurricane events.
The probability distribution of such a count variable
is given by the Binomial distribution
defined as
for
where
and .
The fraction containing factorials on the left hand side
is the number of possible ways successes can happen
out of events, and this can often be surprisingly large.
A binomially distributed variable has expectation
and variance
.
In the limit of large , the binomial distribution is
well approximated by a normal distribution with mean
and variance
.
So for example, if the probability of a hurricane hitting
Florida is , then out of 200 hurricanes, one would
expect a mean of
hurricanes to hit
Florida with a standard deviation of
hurricanes.
The binomial distribution is useful for estimating the
fraction of binary events such as the fraction
of wet days, or the fraction of people voting for a
political party.
Next: Example 3: Poisson distribution
Up: Theoretical discrete distributions
Previous: Example 1: Bernoulli distribution
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David Stephenson
2005-09-30