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In the following, let \(X\) be a Bernoulli random variable with parameter
p = \(p\). Some textbooks also define \(q = 1 - p\), or use
\(\pi\) instead of \(p\).
The Bernoulli probability distribution is widely used to model
binary variables, such as 'failure' and 'success'. The most
typical example is the flip of a coin, when \(p\) is thought as the
probability of flipping a head, and \(q = 1 - p\) is the
probability of flipping a tail.
Support: \(\{0, 1\}\)
Mean: \(p\)
Variance: \(p \cdot (1 - p) = p \cdot q\)
Probability mass function (p.m.f):
$$
P(X = x) = p^x (1 - p)^{1-x} = p^x q^{1-x}
$$
Cumulative distribution function (c.d.f):
$$
P(X \le x) =
\left \{
\begin{array}{ll}
0 & x < 0 \\
1 - p & 0 \leq x < 1 \\
1 & x \geq 1
\end{array}
\right.
$$
Moment generating function (m.g.f):
$$
E(e^{tX}) = (1 - p) + p e^t
$$