Definition:Bernoulli Distribution

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

Then $X$ has the Bernoulli distribution with parameter $p$ :


 * $(1): \quad X$ 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 $p + q = 1$.