0

Is every random variable bijective? Without getting too much into the measure theory behind this, my understanding is that a random variable maps from a sample space $\Omega$ to $\mathbb{R}$ (the random variable $X$ is defined as a function $X: \Omega \rightarrow \mathbb{R}$).

I have yet to come across a case in which each possible outcome in $\Omega$ does not have exactly one value in $\mathbb{R}$, but does this hold true in general? Does one need to delve deeper into the measure theory behind this to appropriately classify the bijectivity of random variables? Thank you!

bjorn
  • 3
  • Constant random variable? – Sangchul Lee Dec 06 '21 at 19:41
  • @SangchulLee Yes, the constant random variable is bijective (always, I think?), but I know that there are other bijective random variables and I'm wondering how that group of random variables could be classified (beyond their bijective property). – bjorn Dec 06 '21 at 19:45
  • Actually the truth is the opposite: A constant random variable can never be a bijection (as a function) from $\Omega$ to $\mathbb{R}$. Even worse, it cannot be an injection if $\Omega$ contains more than one element. – Sangchul Lee Dec 06 '21 at 20:43

0 Answers0