Concept

PMF and CDF of Discrete Uniform Distribution

Definition 89 (Discrete Uniform Distribution (PMF))

Let \(X\) be a discrete random variable that follows a Uniform distribution over the set \(\S\). This means that \(X=x\) has an equally likely chance of being drawn.

Then the probability mass function (PMF) of \(X\) is given by

\[ \P(X=x) = \dfrac{1}{\lvert \S \rvert} \]

More specifically, if \(X\) is a discrete random variable that follows a Uniform distribution over the ordered set \(\S\) where the lower bound is \(a\) and the upper bound is \(b\), then the PMF of \(X\) is given by

\[ \P(X=x) = \dfrac{1}{b-a+1} \qquad \text{for } x = a, a+1, \ldots, b \]

Note:

1. It is non-parametric because there are no parameters associated.

Definition 90 (Discrete Uniform Distribution (CDF))

The cumulative distribution function (CDF) of a discrete random variable \(X\) that follows a Uniform distribution is given by

\[\begin{split} \cdf(x) = \P(X \leq x) = \begin{cases} 0 & \text{if } x < a \\ \dfrac{x-a+1}{b-a+1} & \text{if } a \leq x \leq b \\ 1 & \text{if } x > b \end{cases} \end{split}\]

where \(a\) and \(b\) are the lower and upper bounds of the set \(\S\).

Plotting PMF and CDF of Poisson Distribution

The below plot shows the PMF and its Empirical Histogram distribution for an Uniform distribution with \(a=1\) and \(b=6\), essentially a dice roll.

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fig, axes = plt.subplots(1, 2, figsize=(12, 6), dpi=125)
low, high = 1, 6  # [1, 6] for dice roll
plot_discrete_uniform_pmf(low, high, fig=fig, ax=axes[0])
plot_empirical_discrete_uniform(low, high, fig=fig, ax=axes[1])
plt.show()