Find the distribution of $X_t|Y_t,X_{t-1}$ given $$X_t=\alpha X_{t-1}+V_t$$ $$Y_t=X_t+\sigma_wW_t$$ where $X_1 \sim N(0,1)$, $V_t \stackrel{iid}{\sim} N(0,1)$, and $W_t \stackrel{iid}{\sim} N(0,1)$.
I was trying that $$X_t|X_{t-1} \sim N(\alpha x_{t-1},1)$$ $$Y_t|X_t \sim N(X_t, \sigma^2_w)$$ But how to get the distribution of $X_t|Y_t,X_{t-1}$? Thanks~