Discrete Uniform Distribution

Definition

Discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of \(n\) values has equal probability of \(1 / n\). A random variable \(X\) that follows this distribution is called a discrete random variable.

The discrete uniform distribution itself is inherently non-parametric. However, it is usually represented by an interval \(\left[a, b \right]\), so that \(a\) and \(b\) become the main parameters of the distribution:

\[ \mathcal{U}(a, b). \]

The number of elements in the interval \(\left[a, b \right]\) is \(n = b - a + 1\). The CDF of the discrete uniform distribution is:

\[ F(x; a, b) = \frac{\lfloor x \rfloor - a + 1}{b - a + 1}, \]

for \(\forall x \in \left[a, b \right]\).

Mean and Variance

Recall that the moment-generating function for discrete random variables is:

\[ m_X(t) = \mathbb{E}e^{tX} = \sum_n p_n e^{t x_n}. \]

Then the expected value of the discrete uniform distribution is:

\[ \mathbb{E}X = \left. \frac{d m_X(t)}{dt} \right|_{t = 0} = \frac{1}{n} \frac{(a + b)n}{2} = \frac{a + b}{2}. \]

The variance is the central second moment:

\[ \begin{align} \mathbb{V}\text{ar}X &= \mathbb{E}X^2 - (\mathbb{E}X)^2 \\ &= \left. \frac{d^2 m_X(t)}{dt^2} \right|_{t = 0} - (\mathbb{E}X)^2 \\ &= \frac{n^2 - 1}{12}. \end{align} \]

Usage

The theoretical model for random sampling is the discrete uniform distribution.