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Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2020-07-27 23:17:00 -0700 |
Quantile function of the discrete uniform distribution |
Probability Distributions |
Univariate discrete distributions |
Discrete uniform distribution |
Quantile function |
|
P142 |
duni-qf |
JoramSoch |
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:
$$ \label{eq:duni-qf-qed}
\begin{split}
Q_X(p) &= a + p \cdot (b-a+1) - 1 \\
&= a + pb - pa + p - 1 \\
&= a (1-p) + (b+1) p - 1 ; .
\end{split}
$$