### Artificial IntelligenceAIMA Exercises

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 $\sigma_z^2\rightarrow 0$.

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 $\sigma_z^2\rightarrow 0$.

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