Uniform distribution - Independent Randmo variables?

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Discussion Overview

The discussion revolves around the relationship between uniform distributions and the independence of random variables. Participants explore whether having a uniform distribution implies that the associated random variables are independent, considering various examples and theoretical insights.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions if uniform distribution allows for the conclusion of independence among random variables.
  • Another participant provides an example where two random variables, X and Y, derived from a uniform distribution are not independent, specifically stating Y = 1 - X.
  • A different participant notes that uniformity alone does not imply independence, emphasizing that dependence is a more complex concept that requires additional information.
  • It is mentioned that a test for dependence is based on the relationship E[XY] <> E[X]E[Y], but this condition does not guarantee independence in all cases.
  • One participant suggests that the best approach to assess independence is through the mentioned test, indicating a preference for empirical verification.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are multiple competing views regarding the implications of uniform distribution on independence, with some arguing against the assumption of independence while others provide conditions under which it may be assessed.

Contextual Notes

The discussion highlights the complexity of dependence and independence in the context of uniform distributions, noting that additional parameters or information may be necessary to draw conclusions about the relationship between variables.

karlihnos
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Hi guys,

I have this doubt but i am not sure, if i have an uniform distibution can i conclude that the events or random variables are independent?

Thank you
 
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Could you please explain a little bit more about your data? How many random variables? Which of those variables are uniform...
 
karlihnos said:
Hi guys,

I have this doubt but i am not sure, if i have an uniform distibution can i conclude that the events or random variables are independent?

Thank you

No. For example choose a random variable X from a uniform distribution (0,1) and then let
Y = 1 - X. X and Y are certainly not independent, but both have unform distributions.
 
karlihnos said:
Hi guys,

I have this doubt but i am not sure, if i have an uniform distibution can i conclude that the events or random variables are independent?

Thank you

Generally, distribution without any additional information (e.g. parameters in some dists.) doesn't say anything about dependence. Uniformity of random variable only means, that its realizations are equiprobable, that is [itex]\mathbb{P}[X=x_1] = \mathbb{P}[X=x_2] = ...[/itex] where [itex]X \sim U(a,b)[/itex] and [itex]x_i \in [a,b][/itex]. The notion of (in)dependence is much more tricky. Are you talking about multiple uniformly distributed random variables? A nice example of correlated uniformly distributed rvs gave mathman.
 
karlihnos said:
Hi guys,

I have this doubt but i am not sure, if i have an uniform distibution can i conclude that the events or random variables are independent?

Thank you

The test that will definitely tell if two variables are dependent is if E[XY] <> E[X]E[Y] for two variables X and Y.

The converse is not true though funnily enough: you can show that E[XY] = E[X]E[Y] but still have instances where you have dependent variables, although the case that this happens provides a kind of 'evidence' that they are independent (doesn't mean its conclusive though).
 
Thanks to all. So i think the best way to look for it is the test that mentioned chiro.
 

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