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Example 1: Bernoulli distribution

A Bernoulli (binary) variable is a random variable that can take only the value of either 1 (success) or 0 (failure). Bernoulli variables are commonly used for describing binary processes such as coin tossing, rain/no rain, yes/no decisions etc.. The Bernoulli distribution uses one population parameter $ \pi$ to define the probability of success $ \Pr(X=1)=\pi$ and the probability of failure $ \Pr(X=0)=1-\pi$. This can be written more succinctly as
$\displaystyle \Pr(X=x)$ $\displaystyle =$ $\displaystyle \pi^x(1-\pi)^{1-x}=f(x;\pi)$ (4.1)

where $ x$ takes the value of either 0 or 1. The parameter $ \pi$ completely determines the population distribution and all possible statistics based on $ X$, for example, the population mean is given by $ E(X)=\pi.1+(1-\pi).0=\pi$ and the population variance is given by $ Var(X)=E(X^2)-E(X)^2=\pi.1^2+(1-\pi).0^2-\pi^2=\pi(1-\pi)$. A random variable $ X$ distributed with a Bernoulli distribution is described as $ X\sim {\rm Be}(\pi)$ by statisticians (the $ \sim$ symbol means ``distributed as'').

David Stephenson 2005-09-30