\begin{definition}[Definition:Bernoulli Distribution]
Let $X$ be a discrete random variable on a probability space.
Then $X$ has the '''Bernoulli distribution with parameter $p$''' {{iff}}:
: $(1): \quad X$ 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:
:$\map {p_X} x = p^x \paren {1 - p}^{1 - x}$
\end{definition}