Bernoulli Distribution

The Bernoulli distribution is the simplest non-trivial random variable: a single binary trial that either succeeds or fails. Every more complex discrete distribution — Binomial, Geometric, Negative Binomial — is built directly on top of it.

Definition

A random variable $X$ follows a Bernoulli distribution with parameter $p \in [0, 1]$, written $X \sim \text{Bernoulli}(p)$, if it takes only the values $0$ and $1$ with

$$P(X = 1) = p, \qquad P(X = 0) = 1 - p.$$

The value $1$ is conventionally called success and $0$ is called failure. The single parameter $p$ is the success probability.

PMF in closed form

The two cases can be combined into a single formula:

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

CDF

The cumulative distribution function is piecewise constant:

$$F(x) \coloneqq P(X \le x) = \begin{cases} 0 & x < 0, \\ 1 - p & 0 \le x < 1, \\ 1 & x \ge 1. \end{cases}$$

Mean

The expected value of $X$ follows directly from the definition of expectation for a discrete random variable:

$$E[X] = 0 \cdot P(X = 0) + 1 \cdot P(X = 1) = 0 \cdot (1-p) + 1 \cdot p = p.$$

So $E[X] = p$: the mean is simply the success probability.

Variance

To compute $\text{Var}(X) = E[X^2] - (E[X])^2$, first note that because $X \in \{0, 1\}$ we have $X^2 = X$, so $E[X^2] = E[X] = p$. Therefore

$$\text{Var}(X) = p - p^2 = p(1-p).$$

The variance is maximised at $p = \tfrac{1}{2}$ (maximum uncertainty) and collapses to zero at $p = 0$ or $p = 1$ (the outcome is certain).

Moment generating function

The moment generating function (MGF) of $X$ is

$$M(t) \coloneqq E[e^{tX}] = e^{t \cdot 0}(1-p) + e^{t \cdot 1} p = (1 - p) + p e^t.$$

This compact expression makes it straightforward to derive the MGF of the Binomial distribution by multiplying $n$ independent copies.

Summary

• $X \sim \text{Bernoulli}(p)$ models a single binary trial with success probability $p \in [0,1]$.
• PMF: $P(X = k) = p^k(1-p)^{1-k}$ for $k \in \{0,1\}$.
• Mean: $E[X] = p$.
• Variance: $\text{Var}(X) = p(1-p)$, maximised at $p = \tfrac{1}{2}$.
• MGF: $M(t) = (1-p) + pe^t$.
• The Bernoulli distribution is the atomic building block for the Binomial, Geometric, and Negative Binomial distributions.