# Discrete Uniform distribution¶

## Story¶

A set of discrete outcomes that can be indexed with sequential integers, and each outcome has equal probability, like rolling a fair die.

## Example¶

A monkey can choose from any of $$n$$ bananas with equal probability.

## Parameters¶

The distribution is parametrized by the high and low allowed values, respectively $$y_\mathrm{high}$$ and $$y_\mathrm{low}$$.

## Support¶

The Discrete Uniform distribution is supported on the set of integers ranging from $$y_\mathrm{low}$$ and $$y_\mathrm{high}$$, inclusive.

## Probability mass function¶

\begin{align} f(y ; y_\mathrm{low}, y_\mathrm{high}) = \frac{1}{y_\mathrm{high} - y_\mathrm{low} + 1} \end{align}

## Moments¶

Mean: $$\displaystyle{\frac{y_\mathrm{low} + y_\mathrm{high}}{2}}$$

Variance: $$\displaystyle{\frac{(y_\mathrm{high} - y_\mathrm{low} + 1)^2 - 1}{12}}$$

## Usage¶

Package

Syntax

NumPy

rg.integers(low, high+1)

SciPy

scipy.stats.randint(low, high+1)

Stan

categorical(theta), theta array with all equal values

## Notes¶

• This distribution is not included in Stan. Instead, use a Categorical distribution with equal probailities.

• In SciPy, this distribution is known as scipy.stats.randint. The high parameter is not inclusive; i.e., the set of allowed values includes the low parameter, but not the high. The same is true for rg.integers(), unless you use the endpoint=True keyword argument, in which case the high parameter is inclusive.