1

Let $X$ be a random variable with probability distribution $F$.

To define the question. We have to define to notions.

First, we define the points of support of $X$ as \begin{align} {\rm supp}(X)= \{x: \text{for every open interval } I \ni x \text{ we have that } F(I)>0\} . \end{align}

Now define the quantile functions as for $p \in (0,1)$ \begin{align} Q(p)= \inf \{ x: p \le F(x) \} \end{align}

My question is what is the relationship between ${\rm supp}(X)$ and the range of $Q$? Let's denote the range of $Q$ by $\text{Range}(Q)$.

I think we have the following \begin{align} {\rm supp}(X)= \text{Range}(Q). \end{align}

This is easy to show for discrete distribution with finitely many points and absolutely continuous distributions. Can we show this in general?

I think the key is that $Q(F(X))=X$ almost surely, but I am not entirely sure how to use this to show the desired result formally.

Boby
  • 5,985
  • I don't really understand that supp$(X)$. If $X$ is uniform in $[0,1]$ I would think that it should be $[0,1]$ but it seems that by your definition it is $[0,+\infty)$. The quantile function $Q$ is a generalized inverse of the distribution $F$ which has range $[0,1]$. The domain of $F$ is the range of $X$. I would therefore think that the range of $Q$ is that of $X$. – Kurt G. Nov 19 '22 at 18:30
  • If $X$ is uniform $[0,1]$ and you pick a point $x=2$. Then it is not a support point since any ball around $x=2$ will have probability zero. I am using $F$ as both CDF and probability measure. – Boby Nov 19 '22 at 19:16
  • Ok. So supp$(X)$ is the support of the distribution of $X$. – Kurt G. Nov 19 '22 at 19:28

0 Answers0