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

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