A quick Question on Joint Uniform Distribution

In summary, the question asks if the joint distribution of two uniformly distributed variables on the unit interval will also be uniform on the unit square. It is necessary to have independence between the variables for this to hold true, as the distribution will otherwise be along the diagonal.
  • #1
loveinla
11
0
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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  • #2
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.
 
  • #3
Orodruin said:
Yes, you do need independence. Consider the case X = Y, the resulting distribution is along the diagonal.
Thanks.
 

Related to A quick Question on Joint Uniform Distribution

1. What is a joint uniform distribution?

A joint uniform distribution is a probability distribution that describes the likelihood of two or more continuous random variables occurring simultaneously within a specific range. It is a type of multivariate distribution where all variables have equal probability of occurring within their given range.

2. How is a joint uniform distribution different from a regular uniform distribution?

A regular uniform distribution only describes the probability of one variable occurring within a specific range, while a joint uniform distribution describes the probability of multiple variables occurring simultaneously within their given ranges.

3. How is the probability density function (PDF) calculated for a joint uniform distribution?

The PDF for a joint uniform distribution is calculated by dividing the area of the given range by the total possible area. This can also be represented as a constant value, as all variables have equal probability of occurring within their given range.

4. Can a joint uniform distribution be used to model real-world situations?

Yes, joint uniform distributions can be used to model real-world situations where multiple continuous variables have equal probability of occurring within a specific range. For example, it could be used to describe the probability of two or more weather variables (such as temperature and humidity) occurring simultaneously within a certain range.

5. Are there any limitations to using a joint uniform distribution?

One limitation of using a joint uniform distribution is that it assumes all variables have equal probability of occurring within their given range. This may not accurately reflect real-world situations where certain variables may have different probabilities. Additionally, it can only be used for continuous variables and not discrete variables.

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