# 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, for example $\Img X = \set {a, b}$

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

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

where $0 \le p \le 1$.

That is, the probability mass function is given by:

- $\map {p_X} x = \begin{cases} p & : x = a \\ 1 - p & : x = b \\ 0 & : x \notin \set {a, b} \\ \end{cases}$

If we allow:

- $\Img X = \set {0, 1}$

then we can write:

- $\map {p_X} x = p^x \paren {1 - p}^{1 - x}$

### 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:

- $X \sim \Bernoulli p$

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

- $X \sim \Binomial 1 p$

is often preferred, for notational economy.

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

- $\map {p_X} x = \begin{cases} p & : x = a \\ q & : x = b \\ 0 & : x \notin \set {a, b} \\ \end{cases}$

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.

## Technical Note

The $\LaTeX$ code for \(\Bernoulli {p}\) is `\Bernoulli {p}`

.

When the argument is a single character, it is usual to omit the braces:

`\Bernoulli p`

## Sources

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