# Leibniz rule for double integrals

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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• Physics Forum Leibniz rule double integral.docx
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## Answers and Replies

mathman
Science Advisor
I
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.

mathman
Science Advisor
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

• Physics Forum Leibniz rule double integral.docx
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mathman
Science Advisor
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.

mathman
Science Advisor
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.