I'm stuck with how to deal with the following expected probability expression.
Assume $F(.)$ is a CDF function and $q\in[0,1]$. $$ P(x;q) = q\cdot F(\alpha(x;q)\cdot x)+(1-q)\cdot F((1+\alpha(x;q))\cdot x) $$ where $$\alpha(x;q) = \frac{q\cdot F(\alpha(x;q)\cdot x)}{P(x;q)}\in [0,1]$$
My question: I would like to compare $P(x;q)$ and $F(x)$.
I've tried using implicit function theorem but things suddenly go messy... All I know so far is that $$P(x;1)=P(x;0)=F(x)$$