# Definition:Bernoulli Distribution

## Contents

## Definition

Let $X$ be a discrete random variable on a probability space.

Then $X$ has the **Bernoulli distribution with parameter $p$** if and only if:

- $(1): \quad X$ has exactly two possible values, e.g. $\operatorname{Im} \left({X}\right) = \left\{{a, b}\right\}$

- $(2): \quad \Pr \left({X = a}\right) = p$

- $(3): \quad \Pr \left({X = b}\right) = 1 - p$

where $0 \le p \le 1$.

That is, the probability mass function is given by:

- $p_X \left({x}\right) = \begin{cases} p & : x = a \\ 1 - p & : x = b \\ 0 & : x \notin \left\{{a, b}\right\} \\ \end{cases}$

If we allow:

- $\operatorname{Im} \left({X}\right) = \left\{{0, 1}\right\}$

then we can write:

- $p_X \left({x}\right) = p^x \left({1-p}\right)^{1-x}$

### Success or Failure

The actual values of $a$ and $b$ depends on the particular experiment in question.

However, it is conventional to consider that the outcome whose probability is $p$ is determined to be a **success**, while the other outcome is determined to be a **failure**.

## Also defined as

Some sources insist that $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.

## Notation

This distribution is sometimes written:

- $X \sim \operatorname{Bern} \left({p}\right)$

but as, from Bernoulli Process as Binomial Distribution, the Bernoulli distribution is the same as the binomial distribution where $n = 1$, the notation:

- $X \sim \operatorname{B} \left({1, p}\right)$

is often preferred, for notational economy.

Frequently $q$ is used for $1-p$ in which case the probability mass function is given by:

- $p_X \left({x}\right) = \begin{cases} p & : x = a \\ q & : x = b \\ 0 & : x \notin \left\{{a, b}\right\} \\ \end{cases}$

where $p + q = 1$.

## Also see

- Results about
**the Bernoulli distribution**can be found here.

## Source of Name

This entry was named for Jacob Bernoulli.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.2$: Examples: $(6)$