Bernoulli distribution
Story
The result \(y\) of a single Bernoulli trial is Bernoulli distributed.
Example
Check to see if a given bacterium is competent, given that it has probability \(\theta\) of being competent.
Parameter
The Bernoulli distribution is parametrized by a single value, \(\theta\), the probability that the trial is successful.
Support
The Bernoulli distribution may be nonzero only for \(y = 0\) and \(y = 1\).
Probability mass function
\[\begin{split}\begin{align}
f(y;\theta) = \left\{ \begin{array}{ccc}
1-\theta & & y = 0 \\[0.5em]
\theta & & y = 1.
\end{array}
\right.
\end{align}\end{split}\]
Cumulative distribution function
\[\begin{split}\begin{align}
f(y;\theta) = \left\{ \begin{array}{ccc}
0 & & y < 0 \\[0.5em]
1 - \theta & & 0 \le y < 1 \\[0.5em]
1 & & y \ge 1.
\end{array}
\right.
\end{align}\end{split}\]
Moments
Mean: \(\theta\)
Variance: \(\theta(1-\theta)\)
Usage
Package |
Syntax |
|---|---|
NumPy |
|
SciPy |
|
Distributions.jl |
|
Stan |
|