Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim Beta(\alpha,1)$.
Suppose we have a function of the form: $$ Z_t = \phi(X_t,y_{t-1}) = \lambda X_t + (1-\lambda) min(X_t,y_{t-1}), $$ where $y$ at iteration $t$ is given by: $$ y_t = \frac{1}{t} \sum \limits_{i=1}^{t} Z_i $$ I wish to compute the expected value of $\phi$, that is $\mathbb{E}[\phi(X,y)]$: \begin{align} \mathbb{E}[\phi(X,y)] &= \lambda \mathbb{E}[X] + (1-\lambda) \mathbb{E}[min(X,y)]\\ &= \lambda \frac{\alpha}{\alpha+1} + (1-\lambda) \mathbb{E}[ ?? ] \end{align}
Any help would be appreciated to compute $\mathbb{E}[min(X,y)]$.
Assuming your $X_t$ are iid and also independent of $Z_i$ and having continuous density (this can easily be modified if not):
Hint:
If $F_t,G_t,H_t$ denote the cdf's of $X_t,G_t,R_t$, then $1-H_t(r)=P(R_t>r)=(1-F(t))(1-G(t))$. All your random variables involved here are positive, so $R_t:=\min(X_t,y_{t-1})$ is always positive too. Now recall that for a positive random variable $Y$, $\mathbb{E}[Y]=\int_0^\infty (1-F_Y(t))dt$, where $F_Y$ is the cdf of $Y$. Expanding out $1-H_t(r)$ will simplify things in terms of the expectation of $X_t$.