Consider the following stochastic process:
$$Y_i=c\cdot Y_{i-1}+\varepsilon_i,$$ where $$Y_0\sim \mathcal N(0,\sigma^2)$$ is independent of the white noise $$\varepsilon_i \overset{\text{iid}}\sim \mathcal{N}(0,\sigma^2).$$
(Here iid means independent and identical and $\mathcal N$ denotes the Normal distribution. Also, $\sigma>0$ is a parameter.)
What can be said about the conditional expectation $$\mathbb E(Y_{i+1}\mid Y_1,Y_2,Y_3,\dots,Y_i)$$? What about the conditional Variance $$\mathbb V(Y_{i+1}\mid Y_1,Y_2,Y_3,\dots,Y_i)$$?
My attempt: We can solve the recurrence: For any $n\geq 1$, we have
$$Y_{i+n}=c^n Y_i + E,$$
where $$E=\sum_{i=1}^n c^{n-i} \varepsilon_i \sim\mathcal N\left(0,\sigma^2\frac{1-c^{2n}}{1-c^2}\right).$$
Now I'm stuck.