Appendix C · Study sheet
Statistics
Pointer: statistics.tex
Mean is where you act. Variance is how much you refuse to bet. Covariance is how two errors lean together. A Gaussian is a convenient lie: closed under linear maps, wrecked by wrapping angles and by “the robot is in one of two rooms.”
Independence means the joint factors. Uncorrelated is weaker (covariance zero) and easy to confuse. Bayes: posterior \(\propto\) likelihood \(\times\) prior. That sentence is the localization chapter. Mahalanobis distance \((z-\hat{z})^\top S^{-1}(z-\hat{z})\) is how you tell whether a measurement is surprising under your current ellipse.
Keep
\(\mathbb{E}[x]=\mu\), \(\mathrm{Cov}(x)=\Sigma\). For \(y=Ax+b\), \(\mathrm{Cov}(y)=A\Sigma A^\top\).