1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Obtaining marginal PDFs from joint PDF

  1. Nov 26, 2012 #1
    1. The problem statement, all variables and given/known data
    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.




    2. Relevant 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!



    3. 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
     
  2. jcsd
  3. Nov 27, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    There is nothing wrong; those _are_ the marginal densities! The fact that
    [tex] f_{XY}(x,y) \neq f_X(x) f_Y(y)[/tex] 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.
     
  4. Nov 27, 2012 #3
    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!
     
  5. Nov 27, 2012 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Obtaining marginal PDFs from joint PDF
Loading...