Kalman Filter

Without loss of generality, the probability density function (pdf) of Gaussian distribution can be denoted by function

\begin{align*} N(x, \mu, \sigma) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \end{align*}

where \(\mu\) and \(\sigma\) are the mean and standard deviation repectively. Then, we have

\begin{align*} N(x, \mu_1, \sigma_1) N(x, \mu_2, \sigma_2) = \frac{e^{-\cfrac{(\mu_1-\mu_2)^2}{2(\sigma_1^2+\sigma_2^2)}}}{\sqrt{2\pi(\sigma_1^2+\sigma_2^2)}} N\left(x, \frac{\mu_1\sigma_2^2 + \mu_2\sigma_1^2}{\sigma_1^2 + \sigma_2^2}, \frac{\sigma_1^2\sigma_2^2}{\sigma_1^2+\sigma_2^2}\right) \end{align*}