# Joint Distribution of X & Y: Visualizing Relationships

• Shackleford
In summary, the conversation is about a problem involving joint distributions and the correct limits for integration. The person providing the answer suggests drawing a (u,v) plane and shading the area where u and v are positive and v < u. They also mention the importance of considering whether y > x or not when x and y are both positive. The conversation ends with the person mentioning a new problem and asking whether it should be posted in a new thread or not.
Shackleford
My answer is not really close to the answers provided. They break up the joint distribution into x less than y and y less/equal to x. They have a 1 in their cases. I don't.

http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled-1.png?t=1302832281

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110414_204720-1.jpg?t=1302832297

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Here's a better scan of my work.

http://i111.photobucket.com/albums/n149/camarolt4z28/20110415135619705.png?t=1302894190

http://i111.photobucket.com/albums/n149/camarolt4z28/untitled-2.jpg?t=1302894320

Last edited by a moderator:
You don't have the right limits because when x and y are both positive, it matters whether y > x or not. I would suggest you draw a (u,v) plane and the line v = u. Then shade the area where u and v are positive and v < u. That is where your joint density e-u is nonzero.

Now place a point (x,y) somewhere other than on u = v line. Look at the region described by
u ≤ x and v ≤ y. You only integrate over the part of that that is shaded. And your picture will look different depending whether your point (x,y) is above or below the line v = u. That is why you must have two cases for the (u,v) integrals.

LCKurtz said:
You don't have the right limits because when x and y are both positive, it matters whether y > x or not. I would suggest you draw a (u,v) plane and the line v = u. Then shade the area where u and v are positive and v < u. That is where your joint density e-u is nonzero.

Now place a point (x,y) somewhere other than on u = v line. Look at the region described by
u ≤ x and v ≤ y. You only integrate over the part of that that is shaded. And your picture will look different depending whether your point (x,y) is above or below the line v = u. That is why you must have two cases for the (u,v) integrals.

Crap. I didn't even think about that. That should be no problem. I'll finish it when I get home from work. Thanks.

Okay. I got that. I'm not getting a right answer for a part of the next problem. New thread or should I post it here?

Shackleford said:
Okay. I got that. I'm not getting a right answer for a part of the next problem. New thread or should I post it here?

New questions should be in new threads. If you post it in a thread that already has replies, some helpers likely will skip the thread thinking it is already handled by someone else.

## What is a joint distribution?

A joint distribution is a statistical concept that describes the probability of two or more random variables occurring together. It shows how the values of one variable are related to the values of the other variable.

## Why is it important to visualize the joint distribution of X and Y?

Visualizing the joint distribution of X and Y allows us to see the relationship between the two variables and identify any patterns or trends. This can help us understand how changes in one variable affect the other and make more informed decisions.

## What types of visualizations can be used to represent the joint distribution of X and Y?

Some common visualizations for joint distributions include scatter plots, heat maps, and contour plots. These can help us see the relationship between the two variables and understand the distribution of their values.

## How can the joint distribution of X and Y be used in statistical analysis?

The joint distribution of X and Y can be used to calculate the covariance and correlation between the two variables, which can provide insights into their relationship. It can also be used to make predictions about the values of one variable based on the values of the other.

## Can the joint distribution of X and Y be used to make causal inferences?

No, the joint distribution of X and Y only shows the relationship between the two variables and does not imply causation. To make causal inferences, additional information and analysis are needed.

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