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

$$\Pr(X=x) = \pi^x(1-\pi)^{1-x}=f(x;\pi)$$ (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'').