I was wondering if it is possible to find a density function $p(\cdot|x)$ of a continuous random variable $Y$ such that $\mathbb{E}[Y|x]$ is a non-linear function in terms of $x$, while the (reparameterized) process of generating samples from $Y$ can be represented as a linear function of $x$.
Counter Example (Gaussian): Consider $Y \sim N(x, 1)$, where $N$ denotes the normal distribution with mean $x$ and variance 1. Here, the process of generating a sample from $Y$ can be described as a linear function of $x$, i.e., $x + \epsilon$, where $\epsilon \sim N(0,1)$. On the other hand, the expectation of $Y$, i.e., $\mathbb{E}[Y|x] = x$, is also a linear function of $x$. Therefore, $N(\cdot, 1)$ is not an answer.