Definition – Bernoulli distribution

Contents
1. Definitions
2. Use of the Bernoulli distribution
3. Mean.
4. Variance.

1. Definition

A random variable $X$ is said to follow a Bernoulli distribution if it takes values $0$ and $1$ , and

$p(X=x) = p^x (1-p)^{1-x}$

We write that $X \sim Ber(p)$ .

2. Use of the Bernoulli distribution

We use the Bernoulli distribution to determine the probability of some event $A$ occurring (or not). For example, the probability of rolling a $1$ on a fair $6$ sided dice, is $\frac{1}{6}$ , and so let $X$ be the outcome of whether we roll a $1$ or not. This follows a Bernoulli distribution with $p=\frac{1}{6}$ .

The probability of an event occurring is given by $p(X=1)$ , and the probability of an event not occurring is given by $p(X=0)$ .

3. Mean & Variance

$E(X) = p$
$Var(X) = p(1-p)$

Proof

For the mean, we have $E(X) = \sum_{i=0}^1 p^x (1-p)^{1-x} = 0 + p = p$ . For the variance, consider the identity $Var(X) = E(X^2) -E(X)^2$ . As $0^2 = 0$ and $1^2 = 1$ , we have that $E(X^2) = E(X) = p$ , and so

$Var(X) = p-p^2 = p(1-p)\quad\square$