Joint distribution of independent uniform random variables on [0,1]

[loveinla]

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.

 

[Orodruin]

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.

 

[loveinla]

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

Thanks.