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Sample derivative of double integral

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Problem

Note: the problem is part of a thesis I'm working on.

Define:

> v > 0, where and v are parameters (constant).
v < a < , where a is a parameter (constant).

v1 - variable

F(·) - probability distribution function with support [v, ].
f(·) = F'(·) - probability density function, strictly positive on its support.

G(·) - probability distribution function with support [v, ].
g(·) = G'(·) - probability density function, strictly positive on its support.

I want to derive the expression

,

With respect to v1. Specifically, I want to show that the derivative of I with respect to v1 is positive (the order of integration is first t2 and then t1).
My intuition that this is indeed true is quite simple: I is equal to the expected value of a minimum of two random variable conditional on one variable higher than some cutoff value - v1. As v1 increases and the conditional probability is being adjusted you should end up with a higher expected minimum.

Here's a straight forward derivative of I using the product rule:

I got the first expression by plugging in v1 and thereby getting rid of the external integral. I got the second expression by keeping the double integral and deriving the expression .

I also wrote the first expression (if I'm not wrong!) as:

I'm not sure how to decompose the second expression. I would be very grateful if you could show that the derivative of I is indeed positive. My intuition strongly suggests that it is.

Thank you very much!

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A sample derivative of double integrals are provided.

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See the attached. I think you might make mistakes in the 1st expression you derived.( See the ...

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