X is a random variable so is |X|?

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SUMMARY

In the discussion, participants explore the proof that the absolute value of a random variable |X| is also a random variable. The definition of a random variable is established as a mapping from the sample space (S, F) to the real numbers (R, B(R)). The key argument presented is that since X is measurable and the absolute value function is continuous, the composition of these measurable functions confirms that |X| is also measurable. This conclusion is supported by the fact that the preimage of measurable sets remains within the sigma-algebra F.

PREREQUISITES
  • Understanding of random variables and their definitions
  • Knowledge of measurable functions and sigma-algebras
  • Familiarity with the concepts of continuity in mathematical functions
  • Basic principles of probability theory
NEXT STEPS
  • Study the properties of measurable functions in probability theory
  • Learn about sigma-algebras and their role in defining random variables
  • Explore the concept of continuity in mathematical analysis
  • Investigate the implications of composition of measurable functions
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Mathematicians, statisticians, and students studying probability theory who seek to deepen their understanding of random variables and their properties.

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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