About the joint density function

[simpleeyelid]

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

 

[quadraphonics]

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.

 

[simpleeyelid]

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??

 

[simpleeyelid]

could we say that, since y=2x is a line in R², we have nothing to integrate? sorry that I am so unintuitive...

 

[quadraphonics]

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).

 

[simpleeyelid]

OK, thanks for this, really clear explanation, best wishes for you~