Bernoulli distributions are used to represent events like coin flips
when there is single trial that is either successful or unsuccessful.
The Bernoulli distribution is a special case of the Binomial()
distribution with n = 1.
Details
We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.
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):
$$ E(e^{tX}) = (1 - p) + p e^t $$
See also
Other discrete distributions:
Binomial(),
Categorical(),
Geometric(),
HurdleNegativeBinomial(),
HurdlePoisson(),
HyperGeometric(),
Multinomial(),
NegativeBinomial(),
Poisson(),
PoissonBinomial(),
ZINegativeBinomial(),
ZIPoisson(),
ZTNegativeBinomial(),
ZTPoisson()
Examples
set.seed(27)
X <- Bernoulli(0.7)
X
#> [1] "Bernoulli(p = 0.7)"
mean(X)
#> [1] 0.7
variance(X)
#> [1] 0.21
skewness(X)
#> [1] -0.8728716
kurtosis(X)
#> [1] -1.238095
random(X, 10)
#> [1] 0 1 0 1 1 1 1 1 1 1
pdf(X, 1)
#> [1] 0.7
log_pdf(X, 1)
#> [1] -0.3566749
cdf(X, 0)
#> [1] 0.3
quantile(X, 0.7)
#> [1] 1
cdf(X, quantile(X, 0.7))
#> [1] 1
quantile(X, cdf(X, 0.7))
#> [1] 0