For the AR(1) process defined by $Z_t = aZ_{t-1} + \epsilon_t$, $\epsilon_t \sim Exp(\lambda)$, $a \in (0,1),\lambda >0$, compute $P(Z_t|Z_{t-1})$.
I was only able to compute $E(Z_t|Z_{t-1}) = aZ_{t-1}+1/\lambda$ and $Var(Z_t|Z_{t-1})=1/\lambda^2$, but these are not enough for computing $P(Z_t|Z_{t-1})$. Can someone help me?