Exercise 15.14 [kalman-variance-exercise]

Let us examine the behavior of the variance update in Equation (kalman-univariate-equation) (page kalman-univariate-equation).

1. Plot the value of $\sigma_t^2$ as a function of $t$, given various values for $\sigma_x^2$ and $\sigma_z^2$.

2. Show that the update has a fixed point $\sigma^2$ such that $\sigma_t^2 \rightarrow \sigma^2$ as $t \rightarrow \infty$, and calculate the value of $\sigma^2$.

3. Give a qualitative explanation for what happens as $\sigma_x^2\rightarrow 0$ and as $\sigma_z^2\rightarrow 0$.