Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate discrete distributions ▷ Discrete uniform distribution ▷ Cumulative distribution function

Theorem: Let $X$ be a random variable following a discrete uniform distribution:

\[\label{eq:duni} X \sim \mathcal{U}(a, b) \; .\]

Then, the cumulative distribution function of $X$ is

\[\label{eq:duni-cdf} F_X(x) = \left\{ \begin{array}{rl} 0 \; , & \text{if} \; x < a \\ \frac{\left\lfloor{x}\right\rfloor - a + 1}{b - a + 1} \; , & \text{if} \; a \leq x \leq b \\ 1 \; , & \text{if} \; x > b \; . \end{array} \right.\]

Proof: The probability mass function of the discrete uniform distribution is

\[\label{eq:duni-pmf} \mathcal{U}(x; a, b) = \frac{1}{b-a+1} \quad \text{where} \quad x \in \left\lbrace a, a+1, \ldots, b-1, b \right\rbrace \; .\]

Thus, the cumulative distribution function is:

\[\label{eq:duni-cdf-s1} F_X(x) = \int_{-\infty}^{x} \mathcal{U}(z; a, b) \, \mathrm{d}z\]

From \eqref{eq:duni-pmf}, it follows that the cumulative probability increases step-wise by $1/n$ at each integer between and including $a$ and $b$ where

\[\label{eq:n} n = b - a + 1\]

is the number of integers between and including $a$ and $b$. This can be expressed by noting that

\[\label{eq:duni-cdf-s2b} F_X(x) \overset{\eqref{eq:duni-pmf}}{=} \frac{\left\lfloor{x}\right\rfloor - a + 1}{n}, \; \text{if} \; a \leq x \leq b \; .\]

Also, because $\mathrm{Pr}(X < a) = 0$, we have

\[\label{eq:duni-cdf-s2a} F_X(x) \overset{\eqref{eq:duni-cdf-s1}}{=} \int_{-\infty}^{x} 0 \, \mathrm{d}z = 0, \; \text{if} \; x < a\]

and because $\mathrm{Pr}(X > b) = 0$, we have

\[\label{eq:duni-cdf-s2c} \begin{split} F_X(x) &\overset{\eqref{eq:duni-cdf-s1}}{=} \int_{-\infty}^{x} \mathcal{U}(z; a, b) \, \mathrm{d}z \\ &= \int_{-\infty}^{b} \mathcal{U}(z; a, b) \, \mathrm{d}z + \int_{b}^{x} \mathcal{U}(z; a, b) \, \mathrm{d}z \\ &= F_X(b) + \int_{b}^{x} 0 \, \mathrm{d}z \overset{\eqref{eq:duni-cdf-s2b}}{=} 1 + 0 \\ &= 1, \; \text{if} \; x > b \; . \end{split}\]

This completes the proof.

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Metadata: ID: P141 | shortcut: duni-cdf | author: JoramSoch | date: 2020-07-28, 05:34.