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Example 2: Binomial distribution

Suppose we are interested in counting the number of times $ X$ a Bernoulli event with probability $ \pi$ happens in a fixed number $ n$ 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 $ X\sim Bin(n,\pi)$ defined as

$\displaystyle \Pr(X=m)$ $\displaystyle =$ $\displaystyle \frac{n!}{(n-m)!m!}\pi^m(1-\pi)^{n-m}=f(m;n,\pi)$ (4.2)

for $ m=0,1,\ldots,n$ where $ n!=n(n-1)(n-2)\ldots1$ and $ 0!=1$. The fraction containing factorials on the left hand side is the number of possible ways $ m$ successes can happen out of $ n$ events, and this can often be surprisingly large. A binomially distributed variable has expectation $ E(X)=n\pi$ and variance $ Var(X)=n\pi(1-\pi)$. In the limit of large $ n$, the binomial distribution is well approximated by a normal distribution with mean $ n\pi$ and variance $ n\pi(1-\pi)$. So for example, if the probability of a hurricane hitting Florida is $ \pi=0.1$, then out of 200 hurricanes, one would expect a mean of $ 200\times0.1=20$ hurricanes to hit Florida with a standard deviation of $ \sqrt{200\times0.1\times0.9}=4.2$ hurricanes. The binomial distribution is useful for estimating the fraction of binary events $ X/n$ such as the fraction of wet days, or the fraction of people voting for a political party.


next up previous contents
Next: Example 3: Poisson distribution Up: Theoretical discrete distributions Previous: Example 1: Bernoulli distribution   Contents
David Stephenson 2005-09-30