Generate a simple counterexample

Question

I want to construct a density function $f$ with support $[0,1]$, such that there exists some $q \in (0,1]$ for which the following equation has more than one solutions for $x \in [0,1]$:

$$ x=(1-q)\mathbb{E}(X)+q\mathbb{E}(X|X \leq x) $$

This boils down to:

$$ x=(1-q)\mathbb{E}(X)+q\frac{\int\limits^x_0sf(s)ds}{F(x)} $$

I have tried for several days with various polynomial functions upto cubic but I'm not able to generate such an example. Any help is most appreciated.

Answer 1

Apparently, cubic functions are too smooth for your purpose.

Try $q={1\over2}$ and a uniform distribution over the intervals $[0.00;0.01]\cup[0.30;0.31]\cup[0.69;0.70]\cup[0.99;1.00]$. Then of course $\mathbb E(X)=\frac12$ and ... ... you can see three intersections.