Discussion Overview
The discussion revolves around the existence of joint density functions in probability theory, specifically exploring scenarios where individual random variables have probability density functions (pdfs) but their joint density function does not exist. The scope includes theoretical reasoning and mathematical interpretation.
Discussion Character
- Exploratory, Technical explanation, Conceptual clarification
Main Points Raised
- One participant requests an example of random variables x and y that each have density functions, but whose joint density function does not exist.
- Another participant proposes that if y is defined as a linear transformation of x (specifically, y = 2*x), then while both have pdfs, the joint distribution is concentrated on a set of measure 0, leading to the absence of a joint pdf.
- A participant expresses difficulty in understanding the interpretation of the situation, specifically regarding the derivative of the signed measure.
- Another participant suggests that since y=2x represents a line in R², there is nothing to integrate, which contributes to the non-existence of a joint pdf.
- A later reply reiterates that the joint distribution being nonzero only on a set of measure 0 prevents the existence of a bounded function that integrates to 1, thus confirming the absence of a joint pdf.
Areas of Agreement / Disagreement
Participants appear to agree on the reasoning that the joint distribution being concentrated on a set of measure 0 leads to the non-existence of a joint pdf. However, there is some uncertainty expressed regarding the interpretation of these concepts, particularly in relation to signed measures.
Contextual Notes
There are unresolved aspects regarding the interpretation of signed measures and the implications of having a joint distribution concentrated on a set of measure 0. The discussion does not fully clarify these mathematical nuances.