# Quantile Function

I have seen the definition of quantile function here, which is as follows (slightly modified):

Let $$X$$ be a real-valued non-degenerate random variable with distribution function $$F_X(x)=\mathbb{P}({X\leq x}).$$ The inverse or quantile function $$Q_X$$ of $$F_X$$ is defined as:

$$Q_X(y) = \inf \left\{x\in\mathbb{R}:F_{X}(x)\geq y\right\}$$

where $$0 < y \leq 1$$ , and$$\quad Q_X ( 0 ) = \inf \{ x \in \mathbb { R } : F_X( x ) > 0 \}$$

My question is:

1. Why don't we define the lower endpoint of the range of $$y$$ as $$Q_{X}(0) = -\infty$$

2. Why don't we also define the upper endpoint of the range of $$y$$ of $$Q_{X}(1) = \infty$$

EDITED: deleted "if $$F_{X}(x)<1$$" in "2. Why don't we also define the upper endpoint of the range of y of $$Q_{X}(1) = \infty \;\text{if} \; F_{X}(x)<1$$"

• Does the explanation at en.wikipedia.org/wiki/Quantile_function#Definition help? Commented Nov 24, 2018 at 15:53
• @A. Donda, wikipedia's explanation helps, but it leaves a few subtle points out: the domain of $F_X$ is $[0,1]$, but the domain of $Q_X$ is $(0,1)$ as $0<p<1$. Thus, it does not explicitly say what happens at the end points, $p=0$ and $p=1$. Commented Nov 24, 2018 at 16:16

1. Fixing $$Q_X(0)$$ at $$-\infty$$ would make it completely uninformative when it doesn't need to be. Consider, for example, a finite support distribution such as the uniform between $$a$$ and $$b$$. By the definition of $$Q_X$$ you saw, that would make $$Q_X(0)=a$$, which tells us that the possible values of $$X$$ begin at $$a$$ (which is at least some information). $$Q_X(0)=-\infty$$ would then tell us that the density (or probability if it is not continuous) function of $$X$$ has an infinite tail going left.
• many thanks @Bananin, your explanation definitely made this notation more clear! I guess the notation I saw is intended to cover two possible type of domains - bounded and unbounded. Yes, please, it would be helpful if you could expand 2. and if it is possible to comment why would the domain be $(0,1]$ and not $(0,1)$, in which case we could equally define $Q_X(1)=\text{sup}\{x \in \mathbb{R}: F_X(x)<1\}$ Commented Nov 25, 2018 at 1:22
• For 2. , in the example I gave $Q_X(1)=b$, so $Q_X(1)$ tells us where the possible values of $X$ end on the real line. I think your observation that the definition is meant to cover bounded as well as unbounded domains is the key to questions 1 and 2. With regards to the question about the $sup$, I think your definition would be the same as the one with $inf$ because the sets ${x:F_X(x)<1}$ and ${x:F_X(x)\geq 1}$ are right next to each other. In terms of the domain, $Q_X$ is defined on $(0,1)$. It being defined on $0$ or $1$ depends on the density function being bounded on the left or right. Commented Nov 25, 2018 at 22:28