A quick Question on Joint Uniform Distribution

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SUMMARY

The joint distribution of two random variables X and Y, both uniformly distributed on the unit interval [0, 1], is uniform on the unit square [0, 1]×[0, 1] only if X and Y are independent. If X and Y are not independent, such as in the case where X = Y, the resulting distribution will not cover the entire square uniformly, but will instead be concentrated along the diagonal. Therefore, independence is a necessary condition for achieving a uniform joint distribution in this scenario.

PREREQUISITES
  • Understanding of uniform distribution on the unit interval
  • Knowledge of joint probability distributions
  • Concept of independence in probability theory
  • Familiarity with the unit square [0, 1]×[0, 1]
NEXT STEPS
  • Study the properties of joint distributions in probability theory
  • Explore the implications of independence on joint distributions
  • Learn about conditional distributions and their relationship with independence
  • Investigate examples of dependent random variables and their distributions
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Students and professionals in statistics, data science, and probability theory who are looking to deepen their understanding of joint distributions and the importance of independence in statistical analysis.

loveinla
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Hi, I have a quick question.

If both X and Y are uniformly distributed on the unit interval [0, 1]. Can we prove that the joint distribution of (X, Y) is uniform on the unit square [0, 1]×[0, 1]? Do we need any condition to ensure the result, such as Independence between X and Y?

Thanks.
 
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loveinla said:
Hi, I have a quick question.

If both X and Y are uniformly distributed on the unit interval [0, 1]. Can we prove that the joint distribution of (X, Y) is uniform on the unit square [0, 1]×[0, 1]? Do we need any condition to ensure the result, such as Independence between X and Y?

Thanks.
Yes, you do need independence. Consider the case X = Y, the resulting distribution is along the diagonal.
 
Orodruin said:
Yes, you do need independence. Consider the case X = Y, the resulting distribution is along the diagonal.
Thanks.
 

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