Kalman steady state

Scalar model (13A.31) with $Q = 10^{-5}$, $R = 10^{-3}$, $x_t\equiv1$. (a) Compute $\bar P$, $\bar K$ and the filtered variance $\bar P_{t|t}$ in steady state. (b) Show that the steady-state filter is an EWMA of $y_t$ and give the half-life of its weights. (c) Show that multiplying $Q$, $R$ and $P_0$ by the same constant leaves $b_{t|t}$ unchanged.