 Bernoulli distribution

A Bernoulli random variable records one success as $1$ and one failure as $0$. For $0\le p\le 1$, its probability mass function iswith zero mass elsewhere. The terms success and failure merely name outcomes; success need not be desirable. Any event $A$ produces such a variable through its indicator random variable $B=1_A$, with $p=\mathbb{P}(A)$. A single Bernoulli distribution requires no independence assumption; independence becomes relevant when several trials are combined.

Because $B^2=B$, the expected value and variance follow immediately:For a constructed trial with $p=0.3$, these are $0.3$ and $0.21$. The variance is greatest at $p=1/2$, where it equals $1/4$, since $p(1-p)=1/4-(p-1/2{)}^2$. At $p=0$ or $p=1$, the outcome is certain and the variance vanishes.

The beauty of an indicator random variable is that counting becomes addition: summing such indicators counts successes, and the linearity of expectation gives the expected count even when the indicators are dependent. With independence and a shared $p$, the full count law is the binomial distribution. These models are summarized in Penn State STAT 504, discrete distributions.

A historical anchor is Bernoulli's posthumous Ars conjectandi: its original title page identifies the author and records publication at Basel in 1713. This documents the book's publication, rather than establishing priority for the modern name Bernoulli distribution. See Ars conjectandi, original title page, ETH Library.