Normal Distribution mean What does it mean when we say "consider a normal distribution in variable $x$ whose mean is a linear function $Ay+b$ of second variable $y$"? As per my understanding there is 1 mean of a normal distribution.
 A: Consider a simple linear regression model:
$$y = \beta_0 + \beta_1 \cdot x + \epsilon,$$
$$\epsilon \sim \mathcal{N}(0, \, \sigma^2)$$
Given $x$, the intercept and the slope, the distribution of $y$ is normal. That means that the residual term $\epsilon$ follows a normal distribution with mean $0$. 
It is not wrong to say $y$ follows a normal distribution, as long as you accept a 'mean' that can take on different values depending on $x$:
$$y \sim \mathcal{N}(\beta_0 + \beta_1 \cdot x, \, \sigma^2)$$
(I'm using the more commonly used notation, but in your question, $x$ and $y$ are reversed.)
A: Well for example, it could mean this

$\text{PDF}[\text{NormalDistribution}[2 y+1,1],x]$
In other words, chaining to another variable just adds a dimension to the problem. So, instead of a 2D plot line having a normal distribution, we have have a 3D plot surface with a 2D normal distribution in $y$ and $z$ corresponding to each $x$-axis value, where the mean value is located at $Ay+b$ and $x=Ay+b$. In terms of selecting that distribution for each $x$ that yields $y=\frac{x-b}{A}$.
