Basic Probability

P(A or B) = P(A) + P(B) - P(A and B)

For independent events: P(A|B) = P(A)

For independent events: P(A and B) = P(A) x P(B)

Mutually exclusive: P(A and B) = 0

Histogram Filters

Probability of $Y$ happening given that $X$ has occurred (conditional probability)

$$ P(Y|X) = \frac{P(Y\cap X)}{P(X)} $$

Law of Total Probability - Movement

Bayes Rule - Measurement

Kalman Filters

Measurement Update (Bayes Rule) → product

• New mean $\mu'$:

$$ \mu' = \frac{\frac{\mu}{\sigma^2}+\frac{\nu}{r^2}}{\frac{1}{\sigma^2}+\frac{1}{r^2}}=\frac{r^2\mu + \sigma^2\nu}{r^2+\sigma^2} $$

• New variance $(\sigma^2)'$ - possibly in quiz:

$$ (\sigma^2)' = \frac{1}{\frac{1}{r^2} + \frac{1}{\sigma^2}} $$

Motion Update (LOTP) → convolution

• Updated mean: $\mu'$ = $\mu + u$
• Updated variance: $\sigma^2 = \sigma^2 + r^2$