Define $Y_1$ and $Y_2$ to be two positive and independent random variables, for which the pdf (probability density function) is the same and is given as:
$f(y) = \beta \exp\left(- \beta y\right),\beta>0$.
Suppose that time is slotted, and $t$ ($=0,1,2,...$) is the time index. $Y_1$ is associated with time $t$ and $Y_2$ with time $t+1$.
For $a,b,c>0$, $a>1$ and $c=\frac{\log(a)}{b}$, we define $p(s)$ to be some error probability that is a function of $s$, which is given as follows
\begin{equation}
p(s) = \begin{cases} a \exp\left( - b s \right), & \text{for $s \ge c$} \\ 1, & \text{for $0 < s < c $} \end{cases}
\end{equation}
At time $t$, $s=y_1$. And at time $t+1$, $s=y_1+y_2$.
Let $E_1$ represent the event of having an error at time $t$; the corresponding probability is $p(y_1)$. Also, define $E_2$ to be the event that there is an error at time $t+1$; the corresponding probability is $p(y_1+y_2)$. For $E_2$ to happen (at $t+1$), a necessary condition is that $E_1$ happens (at $t$).
Let $E$ be the event that there is an error at time $t$ and $t+1$. So we have $\mathbb{P}\{ E \}= \mathbb{P}\{ E_1, E_2 \}= \mathbb{P}\{ E_1 \} \mathbb{P}\{ E_2 \mid E_1 \} = p(y_1) p(y_1+y_2)$. So if the realisations $y_1$ and $y_2$ of $Y_1$ and $Y_2$ are known, the (global) error probability is $p(y_1) p(y_1+y_2)$.
I am interested in the case where at $t$ and $t+1$ the realisations of, respectively, $Y_1$ and $Y_2$ are not known. Let $Z= p(Y_1) p(Y_1+Y_2)$. So in this case I want to derive the expected value of $Z$. I also want to derive the CCDF (or CDF) of $Z$.
Here is my first attempt of a solution
\begin{eqnarray}
\mathbb{E} \left\{Z\right\} &=&\int_0^\infty \int_0^\infty p(y_1) p(y_1+y_2) \, f(y_1) f(y_2) \,dy_1 dy_2 \\
&= &\int_0^c \int_0^c \beta \exp\left(- \beta y_1 \right) \beta \exp\left(- \beta y_2 \right) dy_2 dy_1\\
& +&
\int_0^c \int_{c-y_1}^\infty a\exp\left(-b(y_1 +y_2)\right) \beta \exp\left(- \beta y_1 \right) \beta \exp\left(- \beta y_2 \right) dy_2 dy_1 \\
&+& \int_c^\infty \int_{0}^\infty a\exp\left(-b y_1\right) a\exp\left(-b(y_1 +y_2)\right) \beta \exp\left(- \beta y_1 \right) \beta \exp\left(- \beta y_2 \right) dy_2 dy_1.
\end{eqnarray}
Is the above derivation correct?
CCDF $= Pr\left\{Z > z \right\} =$ ?
Please note that if explicit expressions are difficult to derive, I need to at least write these expressions as integrals function of $a$, $b$, $c$, and $\beta$; as done for $E\left\{Z\right\}$.