I have the following equation:
$f(x,y)=Pr(v\xi-y>max\{v-x,0\})$ $ $ $ $ $ $ $ $ $ $ $ $ $\forall x,y$ $ $ $ $ (1)
where $f(x,y)$ is a known function and $f(x,y) \in (0,1)$ , $v$ and $\xi$ are independent random variables with $v\sim F_v$ ($F_v$ is the CDF of $v$) and $\xi \sim F_\xi$ ($F_\xi$ is the CDF of $\xi$ with domain $[1,+\infty$) ). In the above equation (1) $F_v$ is known, whereas $F_\xi$ is unknown.
I want to understand whether there is a unique CDF $F_\xi$ solving equation (1)/ for which equation (1) holds.
How should my proof proceed? Is anyone familiar with similar cases?
When working with probability problems, if you aren't sure how to proceed, a good question to ask is: what if my random variable is…
In your case, suppose $v=0$ a.s. Then $$f(x,y)=\mathbb{P}[{0\cdot\xi-y>\max(v-x,0)}]=\mathbb{P}[-y>\max(v-x,0)]$$ which has no dependence on $\xi$. Consequently it cannot determine $\xi$ uniquely.