Given the rough representation of my scenario below. I am confused about how to proceed with my analysis. I will explain it as follows
Red-point (RP) (Fixed position) is the user which experience interference from the Green point (GP) (Uniformly distributed anywhere in the circle). (Note the GP can be ANYWHERE withing the circle with radius $r_x$, while RP is fixed to the distance $r_c$)
The distance between Red-point and Green point $r$ such that $r \sim U(0,r_c + r_x)$. Since the GP is the Interfering Source. I consider it as the interfering transmitter with Channel distribution $h \sim \exp(1)$
For the sake of brevity, I can't explain the whole process, but the Eq. for interference comes up to be:
$$\mathcal{L}_I(s) = \mathbb{E}[\exp(-shr^{-\alpha})]$$ where $s$ is the constant, $\mathcal{L}(s)$ denotes the Laplace Transform, $h\sim\exp(1)$, $r\sim U(0,r_c+r_x)$, $\alpha$ is a constant. $I$ is the interference that I am trying to calculate.
Now I proceed as follows (I need both suggestions if I am doing right + how to proceed further).
$$\mathbb{E}[I] =\mathcal{L}_I(s) = \mathbb{E}_r[\mathbb{E}_h[\exp(-shr^{-\alpha})]]$$
since $\mathcal{L}_x(s)$ for $x \sim \exp(1) = \frac{1}{1+s}$, so
$$\mathcal{L}_I(s) = \mathbb{E}_r\Big[\frac{1}{1+sr^{-\alpha}}\Big]$$
I know that for $x \sim U(a,b)$, $\mathbb{E}[X] = \frac{a+b}{2}$, but I don't know how to implement it here. Is the following answer correct:
$$\mathcal{L}_I(s) = \frac{1}{1+s(2/(r_c+r_x))^{-\alpha}} $$ $$\mathcal{L}_I(s) = \frac{1}{1+s((r_c+r_x)/)^{\alpha}} $$ $$\mathbb{E}[I] = \frac{1}{1+(\frac{r_c+r_x}{2}s^{1/\alpha})^{\alpha}} $$