#StackBounty: #probability #distributions #random-variable CDF of a function of two random variables

Bounty: 50

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:
$f(y) = beta expleft(- beta yright),beta>0$.
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 expleft( – 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{Zright} &=&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 expleft(- beta y_1 right) beta expleft(- beta y_2 right) dy_2 dy_1\
& +&
int_0^c int_{c-y_1}^infty aexpleft(-b(y_1 +y_2)right) beta expleft(- beta y_1 right) beta expleft(- beta y_2 right) dy_2 dy_1 \
&+& int_c^infty int_{0}^infty aexpleft(-b y_1right) aexpleft(-b(y_1 +y_2)right) beta expleft(- beta y_1 right) beta expleft(- beta y_2 right) dy_2 dy_1.
end{eqnarray}
Is the above derivation correct?

CCDF $= Prleft{Z > z right} =$ ?

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 $Eleft{Zright}$.


Get this bounty!!!

Leave a Reply