Joint Distribution of X & Y: Visualizing Relationships

[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

 

[Shackleford]

Here's a better scan of my work.

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

And the answers provided.

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

 

[LCKurtz]

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.

 

[Shackleford]

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.

 

[Shackleford]

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?

 

[LCKurtz]

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.