1
$\begingroup$

Consider two CDF's $F_1$ and $F_2$ and assume that they satisfy $$ F_1(y-)\leq F_2(y) \quad \quad and \quad \quad F_2(y-)\leq F_1(y) $$ for all $y\in \mathbb{R}$, then it holds that $F_1=F_2$.

I see that $F_1(y)=F_2(y)$ for all $y\in \mathbb{R}$ that is not discontinuity points of $F_1$ and $F_2$, but how do i use this to prove that $F_1(y)=F_2(y)$ for all $y\in \mathbb{R}$?

  • 1
    If y is a discontinuity point of one or both CDFs, you can take a sequence $y_n$ that decrease to y and such that both CDFs are continuous at each $y_n$. This is possible as the point of discontinuity for a CDF are at most countable. Therefore, using right continuity, you have $F_1(y)=\lim_n F_1(y_n)=\lim_n F_2(y_n)=F_2(y)$. – 2017-01-02
  • 0
    Thanks Kolmo, much appreciated. – 2017-01-02

0 Answers 0