# 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 non-joint 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.

## Answer

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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 fool-proof (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