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About the joint density function

  1. Oct 20, 2008 #1
    Could anyone help to give an example

    where in the same proba space, x and y have each the density function, while the joint density function does not exist?

    Thanks in advance,

    Best regards
  2. jcsd
  3. Oct 20, 2008 #2
    Yeah, that's easy. Let x be any r.v. with a pdf. Then, let y = 2*x. Thus, y also has a pdf, but the joint distribution is entirely concentrated on a set of measure 0 (in the x-y plane, that is), and so there is no joint pdf.
  4. Oct 20, 2008 #3
    First, thanks very much indeed..

    but, yes, honestly, not easy for me, I have to think about it...

    how to interpret this by saying the derivative of the signed measure does not exsit??
  5. Oct 20, 2008 #4
    could we say that, since y=2x is a line in R², we have nothing to integrate??? sorry that I am so unintuitive...
  6. Oct 20, 2008 #5
    Basically, yeah. Since the joint distribution is only nonzero on a set of measure 0 (in the xy plane), you can't have a bounded function that integrates to 1 over it (i.e., a joint pdf).
  7. Oct 20, 2008 #6
    OK, thanks for this, really clear explanation, best wishes for you~
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