#gaussian-conditioning

A scalar diffusion has two steps: $x_1 = \sqrt{0.9}\,x_0 + \sqrt{0.1}\,z_1$ and $x_2 = \sqrt{0.8}\,x_1 + \sqrt{0.2}\,z_2$, with independent $z_1, z_2 \sim \mathcal{N}(0, 1)$ (so $\beta_1 = 0.1$, $\beta_2 = 0.2$). What is the variance of $x_1$ given both $x_2$ and $x_0$?