Obtaining marginal PDFs from joint PDF

[JamieL]

Homework Statement

Hi all,
I'm looking at the joint pdf F(x,y) = (8+xy^3)/64) for -1<x<1 and -2<y<2
(A plot of it is here: https://www.wolframalpha.com/input/?i=(8+xy^3)/64+x+from+-1+to+1,+y+from+-2+to+2 ...sorry about the ugly url) and trying to find the marginal PDFs for X and Y.

Homework Equations

I know I want to integrate the joint function with respect to Y and X in order to to get the marginal pdfs for X and Y, respectively. However, I'm running into trouble when I try to set the bounds for these integrals!

The Attempt at a Solution

As far as I can tell, X and Y don't seem to depend on each other in this sense; i.e. for marginal(X) i would have Integral([JointPDF]dy), from -2 to 2, which comes out to 1/2.
(Similarly, integrating with respect to x from -1 to 1 yields 1/4).
When I integrate these from their respective bounds (x from -1 to 1, y from -2 to 2) both come out to 1, as a proper pdf should. However the fact that both are independent of x and y values makes me think something might be wrong...does anyone have any suggestions as to what I might be doing wrong?
Thanks so much!
Jamie

 

[Ray Vickson]

Homework Statement

Hi all,
I'm looking at the joint pdf F(x,y) = (8+xy^3)/64) for -1<x<1 and -2<y<2
(A plot of it is here: https://www.wolframalpha.com/input/?i=(8+xy^3)/64+x+from+-1+to+1,+y+from+-2+to+2 ...sorry about the ugly url) and trying to find the marginal PDFs for X and Y.

Homework Equations

I know I want to integrate the joint function with respect to Y and X in order to to get the marginal pdfs for X and Y, respectively. However, I'm running into trouble when I try to set the bounds for these integrals!

The Attempt at a Solution

As far as I can tell, X and Y don't seem to depend on each other in this sense; i.e. for marginal(X) i would have Integral([JointPDF]dy), from -2 to 2, which comes out to 1/2.
(Similarly, integrating with respect to x from -1 to 1 yields 1/4).
When I integrate these from their respective bounds (x from -1 to 1, y from -2 to 2) both come out to 1, as a proper pdf should. However the fact that both are independent of x and y values makes me think something might be wrong...does anyone have any suggestions as to what I might be doing wrong?
Thanks so much!
Jamie

There is nothing wrong; those _are_ the marginal densities! The fact that
$$f_{XY}(x,y) \neq f_X(x) f_Y(y)$$ just means that the random variables X and Y are dependent.

BTW: we usually try to use lower case letters (such as f) for densities and upper case letters (such as F) for cumulative distribution functions.

 

[JamieL]

Woah - cool!
The more I look at it the more it makes sense, I guess I was just thrown off because I'd never seen an example with a single number before!
If you don't mind my asking, what exactly does this imply? While I understand how to find them, I think I'm slightly by what exactly the marginal PDFs represent?
Thanks again for your help - I really appreciate it!

 

[Ray Vickson]

Woah - cool!
The more I look at it the more it makes sense, I guess I was just thrown off because I'd never seen an example with a single number before!
If you don't mind my asking, what exactly does this imply? While I understand how to find them, I think I'm slightly by what exactly the marginal PDFs represent?
Thanks again for your help - I really appreciate it!

They represent what they always do in such situations: f_X(x) is the density of X when Y is ignored, so the fact that it is a constant means that when X is looked at in isolation it has a uniform distribution.