A quick Question on Joint Uniform Distribution

AI Thread Summary
For two variables X and Y to have a joint uniform distribution on the unit square [0, 1]×[0, 1], they must be independent. If X and Y are not independent, such as when X equals Y, the distribution will not be uniform and will instead concentrate along the diagonal. Independence ensures that the values of X and Y do not influence each other, allowing for a true uniform distribution across the square. Therefore, proving the joint uniformity requires establishing this independence condition. Understanding the relationship between independence and joint distribution is crucial in probability theory.
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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