# #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!!!

This site uses Akismet to reduce spam. Learn how your comment data is processed.