Mean of Piecewise function resting on IID random variables

Question

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$.

I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = \left\{ \begin{array}{l l} \beta x & \quad \text{if $x < \mu$}\\ (1-\beta) + \beta x & \quad \text{if $x \geq \mu$}\\ \end{array} \right. $$ where $\beta \in [0,1]$ and $\mu$ at time $t$ is given by

$$ \mu_t = \frac{1}{t} \sum \limits_{i=1}^{t} Y_i,$$ with $Y_1 = X_1$ and $Y_t = \Phi(X_t, \mu_{t-1}) \quad \forall t >1$.

How can I compute $\mathbb{E}[\Phi (X,\mu^*)]$ for $\mu^*$ (a specific value of $\mu$) ?

$$ \mathbb{E}[\Phi (X,\mu^*)] = \mathbb{E}[(\beta X_t) \mathbf{1}\{X_t<\mu^*\} + (\beta X_t+1-\beta)\mathbf{1}\{X_t\geq \mu^*\}] = .... ? $$

Answer 1

I now have:

\begin{align} \mathbb{E}[\Phi (X,\mu^*)] &= \mathbb{E}[(\beta X_t) \mathbf{1}\{X_t<\mu^*\} + (\beta X_t+1-\beta)\mathbf{1}\{X_t\geq \mu^*\}]\\ &= \mathbb{E}[ (\beta X_t) \mathbf{1}\{X_t<\mu^*\} + (\beta X_t) \mathbf{1}\{X_t\geq \mu^*\} + (1-\beta) \mathbf{1}\{X_t\geq \mu^*\}]\\ &= \beta \mathbb{E}[X] + (1-\beta)\mathbb{E}[\mathbf{1}\{X_t\geq \mu^*\}]\\ &= \beta \mathbb{E}[X] + (1-\beta) \mathbf{P}(X_t\geq \mu^*) \end{align}

Can I write $\mathbf{P}(X_t\geq \mu^*) = \overline{F}(\mu)$ ?