Leibniz rule for double integrals

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Discussion Overview

The discussion revolves around the application of the Leibniz rule for double integrals in the context of differentiating an expected value function with respect to a parameter, specifically $$\beta$$. Participants explore the implications of varying limits of integration and the nature of the integrals involved, including concerns about improper integrals and the boundedness of limits.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a function involving double integrals and seeks to differentiate it with respect to $$\beta$$, referencing the Leibniz rule.
  • Another participant notes that both the upper and lower limits of the integrals depend on $$\beta$$, suggesting that this leads to four integrals instead of two.
  • A participant expresses concern about the nature of the limits, questioning why they are not infinite given the defined state-space of the random variables.
  • There is confusion regarding the attached document and its relevance to the discussion about the limits of integration.
  • One participant highlights that certain regions in the document are unbounded, yet the limits in the expressions after applying the Leibniz rule appear to be bounded, prompting questions about how these limits were derived.
  • Another participant admits to not understanding the context of the graph presented, indicating a lack of clarity in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the limits of integration and the implications of applying the Leibniz rule. There is no consensus on the correct interpretation of the limits or the relationship between the integrals and the state-space defined by the random variables.

Contextual Notes

Participants mention improper integrals and unbounded regions, but the discussion does not resolve how these factors influence the limits of integration or the application of the Leibniz rule.

phoenix2014
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Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is the joint pdf of the two random variables. $$b,K,c_q,\beta$$ are parameters in the problem. I tried using the hints given in a few other sites (http://math.stackexchange.com/quest...le-integral-with-respect-to-upper-limits?rq=1) using Leibniz rule (and extending it to double integrals) but the correct answer eludes me. Additional information (if relevant): the state-space of the two random variables are as follows: $$\xi_1+\xi_2 \geq 2bK+2(1-\beta)c_q, -2bK\leq\xi_1-\xi_2\leq2bK. $$ I have uploaded the entire function from the paper I am referring to, and the regions defined by the random variables as a word document. Please let me know if you need more information. Appreciate any help.
 

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The description for the Liebnitz rule is correct. However in your case, both the upper and lower limits depend on β. Therefore you will have four integrals instead of two.
 
Thank you for the reply. The region defined by the state-space seems to produce improper integrals (like the upper limit of $$\xi_1,\xi_2$$ is infinity). The limits the authors have is only a portion of the region defined. (Please see attached document for region $$\Omega_2$$). Was this by substitution that they get the new limits? Why is not the upper limit infinity? Thank you.
 
What attached document?
 
the one that I have attached again below- the word document "Physics Forum Leibniz rule double integral.docx". Let me know if you are unable to access it. Thanks.
 

Attachments

I looked at the document. I didn't see how it relates to you questions concerning infinity.
 
If you look at regions 6 and 7 they are unbounded and the upper limit of the random variables goes to infinity. However, they have "bounded" limit in their expressions after the Leibniz rule is applied (1-beta)Cq. I was curious why did not have upper limits as infinity. In fact, even region 2 is not bounded but the limits are bounded between (1-beta)Cq and bK+(1-beta)Cq. I was curious how the limits were obtained.
 
I don't know what this is all about. The graph is epsilon_1 versus epsilon_2. I presume there is further context how this relates to the integrals.
 

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