The Bernoulli distribution

Named for the Swiss mathematician Jacob Bernoulli (1654-1705), a Bernoulli trial is a random experiment whose outcome is classified into exactly one of two mutually exclusive, exhaustive categories: success or failure (heads/tails, defective/good item, and so on). Repeating a Bernoulli experiment several independent times, with the probability of success staying the same across repetitions, gives a sequence of Bernoulli trials — any nontrivial experiment can be dichotomised this way.

Definition 11.10 (Bernoulli distribution). Let $X$ be the indicator of a Bernoulli trial's outcome: $X(\text{success})=1$, $X(\text{failure})=0$, with

(equivalently written $f(x)=p^x q^{1-x}$, $q=1-p$). $X$ is a Bernoulli random variable, denoted $X\sim\text{Ber}(p)$, and $f$ its Bernoulli distribution. Its cdf is $F(x)=0$ ($x<0$), $q$ ($0\le x<1$), $1$ ($x\ge1$). …