Joint PDF evaluated over a curve $P_{U,V}$
Hello everyone,
Consider the joint pdf $f_{U,V}(u,v) = f_U(u)f_V(v)$ and a function X of U,V. That is X(U,V). What I want to know is the probability of a certain level curve. That is the probability of X = x. This can also be thought of as the probability of each (U = u, V = v) combination such that X = x.
The paper provided at the link: https://tinyurl.com/yc6c2mdp
Suggests an approach using line integrals. Specifically, the authors write, "Let (U,V) be a random vector with distribution $P_{U,V}$ induced by the PDF $f_{U,V}(u,v) = f_U(u)f_V(v)$. Let X = U+V. For each value of x, there is a corresponding line in the abstract space. Integrating over this line provides the probability per unit length." Multiplying the the arc length of the level curve would then provide the probability of the level curve.
However, while I know the joint PDF describes the probability per unit volume, I don't understand why integrating over the level curve would produce the probability per unit length. Also as in a nonjoint PDF, shouldn't the probability per unit length vary along the curve?
If anyone could provide detailed insight into this statement I would really appreciate it.

You may want to extend the deadline. Allowed time is too short for the level of the question.

Just extended it! Thank you

There is a misunderstanding in your question. Basically, you need to keep reading. I could answer the question you proposed but, as it is, it is simply stemming from a misunderstanding of the paper.

Hi Rage  I have read through the paper a few times but am confused about how to apply the improved method described later on which concludes in equation (4). I want to be able to calculate the probability of a specific level curve X = x given the joint probability distribution. Could you explain how the paper proposes I do this?

That's technically a different problem to the questions you first proposed! I can give you an explanation of the paper and your previous questions but I would ask for a larger bounty.

Hi Rage, I am happy to up the bounty but let me be more specific about what I want to do. I have a joint pdf of two variables (U,V) and a function X which operates on U and V. Let's denote that X(U,V). I want to be able to identify the probability of a specific level curve X = x, using the joint PDF of U,V. That is, find the probability of all values (U,V) which map to x. If you can show me how to do this / explain the paper, I will up the bounty to 45 dollars!

There are several methods but none of them are foolproof (including the one on the paper). All of them have downsides, so I can't guarantee that for the specific function X(u, v) that you have, you will be able to do it What is the function X on your case?

In my case the "function" is actually a system of differential equations unfortunately. That is why I was primarily interested in the method described by the paper (for a specific level curve) because I don't need to worry about the mapping function, just interval in the original PDF

Is this something you want to program or something you want to do theoretically?

I can do it programmatically if I know the technique

I can explain how the paper does it and you would try and see if you can program it.

sounds good. I will up the bounty to 45 dollars for you to accept. A general, detailed explanation would be great. One specific question I have however is in equation (4) what the summation is referring to. I understand the general approach behind equations (1),(2), and (3) but got a bit lost on the abstraction to joint PDFs. Thank you!

The summation appears in both (3) and (4). Do you mean the integral in (4)?

Hi yes, I am confused about the combination of the integral and the summation in equation (4). I understand what the summation refers to in equation (3).

Especially because the summation in (4) doesn't appear in the joint pdf examples given in the last section  only the integral in (4) appears

The summation appears on the second example! I will go over the examples one by one
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