I have attached the image containing the integration that I want to solve.
A and B are constants over here and f(y)f(y) and f(x)f(x) are the probability density functions of random variables y and x respectively. Can someone help me in simplifying this?
There is abuse of notation. The inner integral is calculated w.r.t y variable, yet the inner integral ranges from A-x to y-B. Thus, the variable of the integral y conflicts to the integral range, which is y-B. You should clarify this first.
Dear Prabhupad Bharadwaj
This stack-exchange discussion might be useful, have e a look:
https://math.stackexchange.com/questions/735470/integration-by-simplification
Kind Regards
Qamar Ul Islam
As Ryoungwoo Jang pointed out your notation is quite confusing.
You state that x and y are random variables. You usually would not integrate over random variables. I guess you mean you have the random variables X and Y usually we write the random variables big) and the function f(X=x) is the probability density that X turns out to be x. Then you can integrate over all x (and y respectively).
So to simplify your equation you may pull the x
in "int _(A-x)^(y-B) xf(y)dy" out of the integral since you integrate over y
->
int _0^A ("int _(A-x)^(y-B) f(y)dy) *xf(x)dx
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