Leibniz rule for double integrals

[phoenix2014]

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.

 

[mathman]

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.

 

[phoenix2014]

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]

What attached document?

 

[phoenix2014]

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.

 

[mathman]

I looked at the document. I didn't see how it relates to you questions concerning infinity.

 

[phoenix2014]

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]

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.