Definition:Bernoulli Distribution

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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

In the definition of a Bernoulli distribution, some sources insist that $0 < p < 1$.

However, it can be useful in certain circumstances to include the condition when the outcome is certainty.


The Bernoulli distribution can be 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