Let $x_1$, $x_2$ and $x_3$ be three independent and identically distributed (positive) random variables, for which the Probability Density Function (PDF) is as follows: \begin{equation} f(x) = \alpha \exp\left(- \alpha x\right). \end{equation} Define probability function $p(z)$ as \begin{equation} p(z) = \begin{cases} 1, & \text{when $0 < z < c $} \\ a \exp\left( - b z \right), & \text{when $z \ge c$} \end{cases} \end{equation} Note that $\alpha$, $a$, $b$, and $c$ are all positive constants. Also note that e.g. for $x_1+x_2 \ge c$ we have $p(x_1+x_2)= a \exp\left( - b (x_1 +x_2) \right)$.
Define $y=p(x_1) p(x_1+x_2) p(x_1+x_2+x_3)$.
I am trying to derive the Complementary Cumulative Distribution Function (CCDF) and the expected value of $y$; I am not necessarly looking for explicit expressions, but I need, at least, to write these expressions as integrals function of $\alpha$, $a$, $b$, and $c$.
- Expected value of $y$:
$E\left[y\right]= \int_0^\infty \int_0^\infty \int_0^\infty p(x_1) p(x_1+x_2) p(x_1+x_2+x_3)\, f(x_1) f(x_2) f(x_3) \,dx_1 dx_2 dx_3$ - CCDF $= Pr(y > Y)$ ?