I have the following problem. $\theta^A$ and $\theta^B$ are two i.i.d. random variables, with density f and cumulative F. Support is [0,1]. I look at the following variable: $$b= \begin{cases} \theta^A-\frac{1}{2}\left(\int_0^{\theta^A}F(x)dx + \int_0^{\theta^B}F(x)dx\right) & \text{ if }\theta^B\leq \theta^A\\ \theta^B-\frac{1}{2}\left(\int_0^{\theta^A}F(x)dx + \int_0^{\theta^B}F(x)dx\right) & \text{ if }\theta^B> \theta^A \end{cases}$$
So we have
$$ b=\theta_{(1)}-\frac{1}{2}\left(\int_0^{\theta^A}F(x)dx + \int_0^{\theta^B}F(x)dx\right) $$ where $\theta_{(1)}$ is the first order statistics, i.e., $\theta_{(1)}=max\{\theta^A;\theta^B\}$.
I can't apply the classic formula for convolution. I need to find the probability density of $b$. Any idea? (If needed we can assume that $f$ is uniform.)