Beta distribution
Story
Say you wait for two multistep Poisson processes to arive. The individual steps of each process happen at the same rate, but the first multistep process requires \(\alpha\) steps and the second requires \(\beta\) steps. The fraction of the total waiting time taken by the first process is Beta distributed.
Parameters
There are two parameters, both strictly positive: \(\alpha\) and \(\beta\), defined in the above story.
Support
The Beta distribution has support on the interval [0, 1].
Probability density function
where
is the Beta function.
Moments
Mean: \(\displaystyle{\frac{\alpha}{\alpha + \beta}}\)
Variance: \(\displaystyle{\frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}}\)
Usage
Package 
Syntax 

NumPy 

SciPy 

Stan 

Notes
The story of the Beta distribution is difficult to parse. Most importantly, the Beta distribution allows us to put probabilities on unknown probabilities. It is only defined on \(0 \le \theta \le 1\), and \(\theta\) here can be interpreted as a probability, say of success in a Bernoulli trial.
The case where \(\alpha = \beta = 0\) is not technically a probability distribution because the PDF cannot be normalized. Nonetheless, it is often used as an improper prior, and this prior is known a Haldane prior, names after biologist J. B. S. Haldane. The case where \(\alpha = \beta = 1/2\) is sometimes called a Jeffreys prior.
The Beta distribution may also be parametrized in terms of the location parameter \(\phi\) and concentration \(\kappa\), which are related to \(\alpha\) and \(\beta\) as
The location parameter \(\phi\) is the mean of the distribution and \(\kappa\) is a measure of how broad it is. To convert back to an \((\alpha, \beta)\) parametrization from a \((\phi, \kappa)\) parametrization, use
The mean and variance in terms of \(\phi\) and \(\kappa\) are
Mean: \(\displaystyle{\phi}\)
Variance: \(\displaystyle{\frac{\phi(1\phi)}{1+\kappa}}\).