 Solution

For fixed parameters $(a,b)$, every new error is independent of the previous observations. Put ${\phi }_0(z)=(2\pi {)}^{-1/2}{e}^{-z^2/2}$. Since $X_0=X_1=0$, the first three conditional densities areandIn general,Its density is ${\phi }_0(x_{t+2}-a{x}_{t+1}-b{x}_t)$. The recursion is interpreted from $t=0$, as required to define $X_2$ from the two given initial values. These are parameter-conditional sampling distributions; integrating over the prior would instead give predictive mixtures.