Suppose we have probabiltiy $p$ defined as $ p = (\frac{1}{2}\ln \frac{1}{1-x})\mathbb{1}_{x\leq\alpha})+\mathbb{1}_{x>\alpha}$
I'm trying to find the expectation of $p$, what really bothers me is this vector $\mathbb{1}$ which changes it's behavior based on the value of alpha, aka it could be one when ${x\leq\alpha}$, I'm really unable to resolve it's existence.
My current attempt is like this $\mathbb{E(p)}=$
\begin{cases} \frac{1}{2}\ln \frac{1}{1-\mathbb{E(x)}},& \text{if } x\leq \alpha\\ 1, & \text{otherwise} \end{cases} I'm also not sure which terms I can use to google such a problem.