Are random variables based on the same pmf or pdf always independent?

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Discussion Overview

The discussion revolves around the independence of random variables that share the same probability mass function (pmf) or probability density function (pdf). Participants explore the conditions under which these random variables may or may not be independent, as well as methods for verifying independence.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose that random variables based on the same pmf or pdf are not necessarily independent, citing examples where one variable may be a function of another.
  • One participant provides an example where if X and Y are the same random variable, they are perfectly correlated and thus not independent.
  • Another participant asks about methods to verify the independence of random variables, seeking straightforward approaches or obvious cases.
  • A later reply clarifies that independence can be verified using the condition P(A = a, B = b) = P(A=a)*P(B=b), indicating that this separability defines independence.

Areas of Agreement / Disagreement

Participants generally agree that random variables with the same pmf or pdf do not have to be independent, but multiple competing views on the nature of their independence and verification methods remain unresolved.

Contextual Notes

The discussion does not resolve the complexities of independence verification or the implications of shared distributions on independence.

EdmureTully
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Are they always independent from each other so that you can multiply their E[X] together to form another E[X] with the same distribution and pmf or pdf?
 
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No, in general they need not be independent. In the most extreme case, you may have something like X = some random variable, and Y = X. Then X and Y are 100% correlated, so they are certainly not independent.
 
How do you verify their independence then? Any quick and easy way? Any obvious cases?
 
Hey EdmureTully.

The condition for independence between random variables is P(A = a, B = b) = P(A=a)*P(B=b) where the left hand side is the joint distribution.

If you have this separability, then the two random variables are by definition independent.
 

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