Expectation (Boundaries on the Integral) In a model for hospital room charges X and hospital surgical charges Y for a particular type of hospital admission, the region of probability (after scaling units) is 0 ≤ y ≤ 2x+1 ≤ 3

The joint density function of X and Y is f(x, y) = 0.3(x + y) 

Find the expected excess of surgical charges over room charges for an admission.



Am I solving a double integral here of 0.3(x+y) dxdy? Cause Im having trouble finding the boundaries on the integral?
You have the joint support of: $0 ≤ y ≤ 2x+1 ≤ 3$

That is $(0\leq y \leq 3) \times (\frac{y-1}{2}\leq x\leq \frac{3-1}{2})$, which gives you

$$\int_0^3 \int_{(y-1)/2}^{1} (...)\; 0.3 (x+y)\operatorname d x\operatorname d y$$

Alternatively we could use :

$$\int_{-1/2}^1 \int_{0}^{2x+1} (...)\; 0.3 (x+y)\operatorname d y\operatorname d x$$
You have the joint support of: $0 ≤ y ≤ 2x+1 ≤ 3$

That is $(0\leq y \leq 3) \times (\frac{y-1}{2}\leq x\leq \frac{3-1}{2})$, which gives you

$$\int_0^3 \int_{(y-1)/2}^{1} (...)\; 0.3 (x+y)\operatorname d x\operatorname d y$$

Alternatively we could use :

$$\int_{-1/2}^1 \int_{0}^{2x+1} (...)\; 0.3 (x+y)\operatorname d y\operatorname d x$$