X is a random variable so is |X|?

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Discussion Overview

The discussion revolves around the question of whether the absolute value of a random variable, |X|, is also a random variable. Participants explore definitions, properties of measurability, and the implications of these concepts in the context of probability theory.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant asks for suggestions on how to prove that |X| is a random variable given that X is a random variable.
  • Another participant requests clarification on the definition of a random variable.
  • A definition is provided, indicating that a random variable is a mapping from a probability space (S,F) to the real numbers (R, B(R)).
  • It is noted that since X is measurable and the absolute value function is continuous, |X| is also measurable, as the composition of measurable functions is measurable.
  • A participant suggests that using the pre-image of sets under X and |X| could demonstrate that |X| is measurable, referencing the properties of Borel sets.
  • One participant mentions that their math professor confirmed the reasoning about measurability, suggesting they may have initially overthought the problem.

Areas of Agreement / Disagreement

Participants generally agree on the properties of measurability and the implications for |X|, but there is no explicit consensus on a formal proof or resolution of the initial question posed.

Contextual Notes

The discussion does not resolve the formal proof of |X| being a random variable, and assumptions about the definitions and properties of measurability are not fully explored.

BoogieE
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Howdy guys. Given that X is a random variable how would you prove |X| to be one too? Thanks for any suggestions!
 
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What is your definition of random variable?
 
Just mapping from (S,F) to (R, B(R))
 
X is measurable, | | is continuous hence measurable. And the composition of measurables is measurable.
 
I am expecting that from the fact that X(-1)(G) = {w in S such that X(w) is in G for all G in B(R) } is in F you can somehow show that |X|(-1)(G') = {w in S such that |X|(w) is in G' for all G' in B(R)} is also if F
 
micromass said:
X is measurable, | | is continuous hence measurable. And the composition of measurables is measurable.
I asked my math professor and she said this is ok. I probably overthought the problem. Thank you very much!
 

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