About the joint density function

AI Thread Summary
In the discussion, an example is provided where two random variables, x and y, each have their own probability density functions (pdfs), but their joint density function does not exist. This occurs when y is defined as a linear transformation of x, specifically y = 2*x, resulting in the joint distribution being concentrated on a set of measure 0 in the x-y plane. Consequently, there is no joint pdf since it cannot integrate to 1 over a set of measure 0. The conversation emphasizes the difficulty in interpreting this concept, particularly regarding the existence of a derivative of the signed measure. Overall, the explanation clarifies the relationship between individual and joint distributions in probability theory.
simpleeyelid
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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
 
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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.
 
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??
 
could we say that, since y=2x is a line in R², we have nothing to integrate? sorry that I am so unintuitive...
 
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).
 
OK, thanks for this, really clear explanation, best wishes for you~
 

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