Bernoulli

Definition

A discrete probability distribution for a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q=1-p$. It represents a single experiment with a binary outcome (success/failure). It is denoted as $X\sim \text{Bernoulli}(p)$ or $\text{Be}(p)$.

Probability Mass Function (PMF)

The probability function describes the likelihood of the two possible outcomes $k\in \{0,1\}$.

$$P(X=k)=\{\begin{array}{ll}q=1-p& \text{for }k=0\\ p& \text{for }k=1\end{array}$$

This can also be expressed analytically as:

$$P(X=k)=p^k(1-p{)}^{1-k}\text{for }k\in \{0,1\}$$

Cumulative Distribution Function (CDF)

The CDF is a step function.

$$F(k)=\{\begin{array}{ll}0& \text{for }k<0\\ 1-p& \text{for }0\le k<1\\ 1& \text{for }k\ge 1\end{array}$$

Key Statistics

Expectation (Mean):

$$E[X]=p$$

Variance:

$$\text{Var}(X)=p(1-p)=pq$$

Moment Generating Function:

$$M_X(t)=(1-p)+p{e}^t$$

Relation to Binomial Distribution

The Bernoulli distribution is a special case of the Binomial distribution where the number of trials $n=1$. Conversely, a Binomial distribution $B(n,p)$ is the sum of $n$ independent and identically distributed Bernoulli trials.