Current-controlled source or CCS

Let’s consider the same circuit but with the current source, that is function of input current.

Figure 1.

Let’s assume that $I_{out}=f\left(I_{in}\right)$. At the same time $I_{out}=f\left(I_{in}\right)=\frac{V_{out}}{R_{out}}$.

Let’s represent $I_{out}=–\alpha I_{in}$, then $V_{out}=–\alpha I_{in}{R}_{out}$.

Here $I_{in}=\frac{v_{in}}{R_{in}}$, then $V_{out}=–\alpha V_{in}\frac{R_{out}}{R_{in}}$.