In AR(1), why is $X_i|X_{i-1}=x_{i-1}\sim N(\alpha x_{i-1},\sigma^2)$? The AR(1) model starting with $X_0=0$ is
$$X_i=\alpha X_{i-1}+\epsilon_i, \ i=1,...,n, \ -1<\alpha<1$$
where $\epsilon_i\sim N(0,\sigma^2)$ are independent error terms.
Why then is $$X_i|X_{i-1}=x_{i-1}\sim N(\alpha x_{i-1},\sigma^2)?$$
 A: $X_i$ conditioned on $X_{i-1}=x_{i-1}$ implies $$X_i=\alpha x_{i-1}+\epsilon_{i},$$ i.e. a normal random variable $\epsilon_i$ adding a constant $\alpha x_{i-1}$. So $X_i\mid X_{i-1}=x_{i-1}$ is still a normal random variable, such that $$\operatorname{E}[X_i\mid X_{i-1}=x_{i-1}]=\operatorname{E}[\alpha x_{i-1}+\epsilon_{i}]=\alpha x_{i-1}+\operatorname{E}[\epsilon_i]=\alpha x_{i-1},$$ and $$\operatorname{Var}(X_i\mid X_{i-1}=x_{i-1})=\operatorname{Var}(\alpha x_{i-1}+\epsilon_{i})=\operatorname{Var}(\epsilon_{i})=\sigma^2.$$
A: I think the notation is confusing you here.
$X_i |X_{i-1} = x_{i-1} \sim N(\alpha x_{i-1}, \sigma^2)$ is another way of writing $X_i |(X_{i-1} = x_{i-1}) \sim N(\alpha x_{i-1}, \sigma^2)$. The equality is not meant to give a value for $X_{i}|X_{i-1}$, but meant to define a conditional distribution for when $X_{i-1}$ takes the realization $x_{i-1}$.
The notation $X_i |X_{i-1} = x_{i-1} \sim N(\alpha x_{i-1}, \sigma^2)$ means that conditional on the random variable $X_{i-1}$ taking value $x_{i-1}$, the distribution of $X_{i}$ is a normal distribution with mean $\alpha x_{i-1}$ and variance $\sigma^2$.
