Discussion Overview
The discussion centers on the definitions of random variables, particularly whether a random variable can be equivalently defined as a subset of ℝ associated with its cumulative distribution function (CDF) rather than as a function from a sample space to the reals. The scope includes theoretical considerations and clarifications regarding the nature of random variables and their relationships to probability distributions.
Discussion Character
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant suggests that a random variable could be defined as a subset of ℝ associated with its CDF, based on a statement from a Wikipedia article.
- Another participant counters that multiple distinct random variables can correspond to the same CDF, emphasizing that they are not the same function despite sharing a probability distribution.
- A third participant critiques the Wikipedia article for containing contradictions, illustrating their point with an example involving two different questionnaires that could yield random variables with the same distribution but defined on different probability spaces.
- A later reply mentions that a graduate course in probability would provide a more precise definition involving sigma algebras and Borel spaces, suggesting a more rigorous framework for understanding random variables.
Areas of Agreement / Disagreement
Participants express disagreement regarding the equivalence of defining random variables in terms of CDFs versus as functions from sample spaces. There is no consensus on the definitions, and the discussion remains unresolved.
Contextual Notes
Participants highlight the importance of the underlying probability space in defining random variables, indicating that definitions may depend on specific contexts and assumptions.