## Introduction

In the theory of probability and statistics, a **Bernoulli trial** or **Bernoulli Experiment** is a random experiment with exactly two mutually exclusive outcomes, “Success” and “Failure” with the probability of success remains same every time the experiment is conducted. The name **Bernoulli trial** or **Bernoulli distribution** named after a Swiss scientist **Jacob Bernoulli**.

## Bernoulli Trials

Trials of random experiment are called Bernoulli trials, if they satisfy the following conditions

- Each trial of a random experiment has two possible outcomes (like “Success” and “Failures”).
- The trials are independent. That is the outcome of one trial has no influence on the outcome of another trial.
- The probability of success remains constant from one trial to another.

Examples of Bernoulli trial are

- Tossing of a coin (“Head” or “Tail”),
- Sex of newborn baby (“Male” or “Female”),
- Answer to true/false question (“True” or “False”).

## Bernoulli Distribution

Consider a random experiment having two possible outcomes, namely, “Success” (S) and “Failure” (F) with respective probabilities $p$ and $q$. The outcomes success and failures are mutually exclusive. Such a trial is called **Bernoulli trial**.

The probability distribution of the random variable $X$ representing the number of success obtained in a Bernoulli trial is called **Bernoulli distribution**. Thus the random variable $X$ takes the value 0 and 1 with respective probabilities $q$ and $p$, i.e.,
`$$ \begin{equation*} P(F) = P(X=0) = q, \text{ and } P(S) = P(X=1) = p. \end{equation*} $$`

## Definition

The discrete random variable $X$ is said to have Bernoulli distribution if its probability mass function is given by
`$$ \begin{equation*} P(X=x) = p^x q^{1-x}, \; x=0,1; 0<p,q<1; q=1-p. \end{equation*} $$`

Here

- $P(X=x)\geq 0$ for all $x$.
- $\sum_{x} P(X=x) = P(X=0) + P(X=1) = q+p =1$.

Hence $P(X=x)$ is a legitimate probability mass function.

## Key features of Bernoulli's Distribution

- There are only two outcomes for a random experiment like success ($S$) and failure ($F$).
- The outcomes are mutually exclusive.
- The probability of success is $p$.
- The random variable $X$ is the total number of success.

## Graph of Bernoulli Distribution

## Mean of Bernoulli Distribution

The mean (expected value) of Bernoulli random variable $X$ is $E(X) = p$.

### Proof

The expected value of Bernoulli random variable $X$ is
`$$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) E(X) &=& \sum_{x=0}^1 x P(X=x) \\ &=& 0\times P(X=0) + 1\times P(X=1)\\ &=& 0\times q + 1\times p = p. \end{eqnarray*} $$`

## Variance of Bernoulli Distribution

The variance of Bernoulli random variable $X$ is $V(X) = pq$.

### Proof

Variance of random variable $X$ is given by
`$$ V(X) = E(X^2) - [E(X)]^2. $$`

Let us find the expected value of $X^2$.
`$$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) E(X^2) &=& \sum_{x=0}^1 x^2 P(X=x) \\ &=& 0\times P(X=0) + 1\times P(X=1)\\ &=& 0\times q + 1\times p = p. \end{eqnarray*} $$`

Thus variance of $X$ is
`$$ \begin{eqnarray*} V(X) &=& E(X^2)-[E(x)]^2\\ &=& p-p^2 = p(1-p)=pq. \end{eqnarray*} $$`

## Moment Generating Function of Bernoulli Distribution

The moment generating function (M.G.F.) of Bernoulli distribution is given by $M_X(t) = (q + pe^t)$ for $t\in R$.

### Proof

The moment generating function of Bernoulli random variable $X$ is
`$$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) M_X(t) &=& E(e^{tX}) \\ &=& \sum_{x=0}^1 e^{tx}P(X=x)\\ &=& e^0 P(X=0) + e^tP(X=1)\\ &=& q+pe^t. \end{eqnarray*} $$`

## Probability Generating Function of Bernoulli Distribution

The probability generating function (P.G.F.) of Bernoulli distribution is given by $P_X(t) = q+pt$, $t\in R$.

### Proof

The probability generating function of Bernoulli random variable $X$ is given by
`$$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) P_X(t) &=& E(t^{X}) \\ &=& \sum_{x=0}^1 t^xP(X=x)\\ &=& t^0 P(X=0) + t^1P(X=1)\\ &=& q+pt. \end{eqnarray*} $$`

## Characteristic Function of Bernoulli Distribution

The Characteristic function of Bernoulli distribution is given by $\phi_X(t) = (q + pe^{it})$.

### Proof

The characteristic function of Bernoulli random variable $X$ is
`$$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \phi_X(t) &=& E(e^{itX}) \\ &=& \sum_{x=0}^1 e^{itx}P(X=x)\\ &=& e^0 P(X=0) + e^{it}P(X=1)\\ &=& q+pe^{it}. \end{eqnarray*} $$`