I did not find any reference over the stack community and internet, but hopefully you might help me. I am trying to compute the expectation (over time $t$) of the following (composite) indicator function:
$\mathbb{E}\left[\mathbb{1}\{u_{t} \leq \mathbb{1}\{X_{t} \geq x\}k\}\right]$,
where $u_{t}$ are independent random draws from from a $U[0,1]$ in every period $t$, $\mathbb{1}\{X_{t} \geq x\}$ is a standard indicator function that returns $1$ if the condition inside the curly bracket is true and $0$ otherwise, and $k$ is a constant.
I am wondering if the expression above is equivalent to the one below
$\mathbb{E}\left[\mathbb{1}\{u_{t} \leq \mathbb{E}\left[\mathbb{1}\{X_{t} \geq x\}\right]k\}\right]$.
If it is, I can carry forward the whole calculation. I just need to show this first step.
Any suggestion is kindly appreciated.
The expressions are equal (not sure if I'd call them "equivalent") if $k\le1$. If $k\gt1$, the second expression will generally be greater, since taking the expectation will tend to avoid the clipping.
In detail: Let $k\in[0,1]$, and $p_1=\mathbb P(X_t\ge x)$ and $p_0=\mathbb P(X_t\lt x)$; then
$$ \mathbb{E}\left[\mathbb{1}\{u_{t} \leq \mathbb{1}\{X_{t} \geq x\}k\}\right]=p_0\mathbb1\{u_t\le0\}+p_1\mathbb1\{u_t\le k\}=p_1k $$
and
$$ \mathbb{E}\left[\mathbb{1}\{u_{t} \leq \mathbb{E}\left[\mathbb{1}\{X_{t} \geq x\}\right]k\}\right]=\mathbb{E}\left[\mathbb{1}\{u_{t} \leq p_0\cdot0+p_1k\}\right]=\mathbb{E}\left[\mathbb{1}\{u_{t} \leq p_1k\}\right]=p_1k\;. $$