How to visualize joint uniform distribution

In summary, the joint density function for X and Y is uniform over the triangular region defined by 0<=x<=y<=L, with a triangular prism shape. The expected values for the squares of X and Y can be calculated by integrating, with E[X^2] = L^2/6 and E[Y^2] = L^2/2 when L = 1.
  • #1
Mad Scientists
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0
Lets say you have X and Y, where the joint density function for X and Y is uniform over the region defined by 0<=x<=y<=L, where L is some positive constant.

The question asks for the expected value of the squares of X and Y.

I am having trouble visualizing what such a distribution would look like. Apparently it is triangular shaped... but I do not see it. Can anyone help?

Thanks in advance...
:confused:
 
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  • #2
Suppose L = 1. Then the distribution is defined over the triangle above the y = x line in the unit square. The frequency is marked on the Z axis and is constant over the whole triangular area. So, you have a triangular prism; its volume = 1 by definition.
 
  • #3
just integrate,
you will get:
[tex]E[X^2]=L^2/6[/tex]
[tex]E[Y^2]=L^2/2[/tex]
 

What is a joint uniform distribution?

A joint uniform distribution is a type of probability distribution where two or more random variables have equal probability of occurring within a specified range. It is often used to model situations where there is no preference or bias towards any particular outcome.

How do I visualize a joint uniform distribution?

One way to visualize a joint uniform distribution is to create a scatter plot, where the x and y axes represent the two random variables. Each data point on the plot will have equal probability, and the points should be evenly distributed within the specified range.

What is the purpose of visualizing a joint uniform distribution?

Visualizing a joint uniform distribution can help to understand the relationship between two or more variables and identify any patterns or trends. It can also aid in making predictions and analyzing data in statistical analyses.

What are some common applications of joint uniform distributions?

Joint uniform distributions are commonly used in fields such as economics, physics, and engineering to model random processes and phenomena. They can also be used in simulations and machine learning algorithms.

How can I generate a joint uniform distribution in statistical software?

Many statistical software packages, such as R or Python, have built-in functions to generate joint uniform distributions. These functions typically require specifying the range of the variables and the number of data points to be generated.

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