Binomial distribution
Story
We perform \(N\) Bernoulli trials, each with probability \(\theta\) of success. The number of successes, \(n\), is Binomially distributed.
Example
Distribution of plasmids between daughter cells in cell division. Each of the \(N\) plasmids as a chance \(\theta\) of being in daughter cell 1 (“success”). The number of plasmids, \(n\), in daughter cell 1 is Binomially distributed.
Parameters
There are two parameters: the probability \(\theta\) of success for each Bernoulli trial, and the number of trials, \(N\).
Support
The Binomial distribution is supported on the set of nonnegative integers less than or equal to \(N\).
Probability mass function
\[\begin{split}\begin{align}
f(n;N,\theta) = \begin{pmatrix}
N \\
n
\end{pmatrix}
\theta^n (1\theta)^{Nn}.
\end{align}\end{split}\]
Moments
Mean: \(N\theta\)
Variance: \(N\theta(1\theta)\)
Usage
Package 
Syntax 

NumPy 

SciPy 

Stan 
