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Let $X$ be a random variable and $f$ a non-decreasing function on the range of $X$. If $Y=f(X)$, then $$\{X\le q\}\subset\{Y\le f(q)\} \quad\quad\text{and}\quad\quad \{X<q\}\supset\{Y<f(q)\}$$ for any $q\in\mathbb R$.

How can one see these inclusions?

I've tried the following approach. We can write
$$\{X\le q\}=X^{-1}\left((-\infty,q]\right)\quad\quad\text{and}\quad\quad \{Y\le f(q)\}=X^{-1}(f^{-1}((-\infty,f(q)]))$$ Now if I could show that $(-\infty,q]\subset f^{-1}((-\infty,f(q)])$, then the first inclusion would follow, but I am struggling to see how.

Phil-ZXX
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1 Answers1

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  1. $ x \le q \Rightarrow f(x) \le f(q)$ thus $\omega \in [X(\omega)\le q] \Rightarrow \omega \in [Y(\omega)\le f(q)]$.
  2. $f(x) <f(q) \Rightarrow x<q$ thus $\omega \in [Y(\omega)<f(q)] \Rightarrow \omega \in [X(\omega)<q]$.
Sergio Parreiras
  • 3,813