The Bernoulli Probability Distribution

Chapter 4. Probability Distributions: Discrete Probability Distributions

The Bernoulli Probability Distribution

Bernoulli Trial
A Bernoulli trial is a random experiment with two possible outcomes: success or failure.

$\Omega=\{\text{success}, \text{ failure}\}$ Typically, a success is encoded as a ' $1$ ' and a failure as a ' $0$ '.

In a Bernoulli trial, the probability of observing a success is denoted by $p$ and the probability of failure is denoted by $q$ :

$\begin{array}{lcrcl} \mathbb{P}(\text{success})\,=\,p&\phantom{Batman}&\mathbb{P}(\text{failure})&=&q\\\\ &&&=&1-p \end{array}$

Bernoulli Random Variable
Let $X$ be a random variable that takes on a value of $1$ with probability $p$ and a value of $0$ with probability $1-p$ , then $X$ is a Bernoulli random variable.

We say that $X$ is Bernoulli distributed with parameter $p$ and write this as:

$X \sim Bernoulli(p)$

GeoGebra

Suppose we roll a regular six-sided die and define a success as rolling an even number.

Let $X$ be a random variable that takes on a value of $1$ when we roll an even number and a value $0$ when we roll an uneven number.

Then $X$ is Bernoulli distributed with $p=\mathbb{P}(\text{Roll an even number}) = 0.5$ :

$X\sim Bernoulli(0.5)$

$\text{}$

Let $X$ be a Bernoulli distributed random variable with parameter $p$ .

Then the expected value of $X$ calculated with the following formula:

$\mu = p$

The variance of $X$ is calculated with the following formula:

$\sigma^2 = p \cdot (1-p)$

And the standard deviation of $X$ is calculated with the following formula:

$\sigma = \sqrt{p \cdot (1-p)}$

Let $X$ be a Bernoulli distributed random variable with parameter $p=0.75$ .

Calculate the expected value of $X$ .

$\mu = 0.75$

The expected value of a Bernoulli random variable $X\sim Bernoulli(p)$ is calculated as follows:

$\begin{array}{rcl} \mu&=&p\\\\ &=& 0.75 \end{array}$

New example

The Bernoulli distribution serves as the foundation for two other discrete probability distributions:

the binomial distribution
the geometric distribution