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

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

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

Then, the quantile function of $X$ is

\[\label{eq:duni-qf} Q_X(p) = \left\{ \begin{array}{rl} -\infty \; , & \text{if} \; p = 0 \\ a (1-p) + (b+1) p - 1 \; , & \text{when} \; p \in \left\lbrace \frac{1}{n}, \frac{2}{n}, \ldots, \frac{b-a}{n}, 1 \right\rbrace \; . \end{array} \right.\]

with $n = b - a + 1$.

Proof: The cumulative distribution function of the discrete uniform distribution is:

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

The quantile function $Q_X(p)$ is defined as the smallest $x$, such that $F_X(x) = p$:

\[\label{eq:qf} Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .\]

Because the CDF only returns multiples of $1/n$ with $n = b - a + 1$, the quantile function is only defined for such values. First, we have $Q_X(p) = -\infty$, if $p = 0$. Second, since the cumulative probability increases step-wise by $1/n$ at each integer between and including $a$ and $b$, the minimum $x$ at which

\[\label{eq:duni-cdf-p} F_X(x) = \frac{c}{n} \quad \text{where} \quad c \in \left\lbrace 1, \ldots, n \right\rbrace\]

is given by

\[\label{eq:duni-qf-p} Q_X\left( \frac{c}{n} \right) = a + \frac{c}{n} \cdot n - 1 \; .\]

Substituting $p = c/n$ and $n = b - a + 1$, we can finally show:

\[\label{eq:duni-qf-qed} \begin{split} Q_X(p) &= a + p \cdot (b-a+1) - 1 \\ &= a + pb - pa + p - 1 \\ &= a (1-p) + (b+1) p - 1 \; . \end{split}\]
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Metadata: ID: P142 | shortcut: duni-qf | author: JoramSoch | date: 2020-07-28, 06:17.