Gaussian Distribution

Gaussian Variable

Learned these properties from the SLAM textbook and Kalman Filter book.

Univariate Case

We define $x\sim N({\mu }_x,{\sigma }_x^2),y\sim N({\mu }_y,{\sigma }_y^2)$

Sum of Gaussians

If we have two independent gaussian variables, their sum is still gaussian, i.e. $x+y\sim N({\mu }_x+{\mu }_y,{\sigma }_x^2+{\sigma }_y^2)$ Other notation: $\mu ={\mu }_1+{\mu }_2$ ${\sigma }^2={\sigma }_1^2+{\sigma }_2^2$

Product with Constant Factor

If we multiple $x$ with a constant factor $a$, then $ax\sim N(a{\mu }_x,a^2{\sigma }_x^2)$

Product of Gaussians

A product of two gaussian distributions is not gaussian. However, it is proportional to a gaussian $N(\mu ,{\sigma }^2)$, where:

$\mu =\frac{{\sigma }_1^2{\mu }_2+{\sigma }_2^2{\mu }_1}{{\sigma }_1^2+{\sigma }_2^2}$ ${\sigma }^2=\frac{{\sigma }_1^2{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$

This is taken from the kalman filter book.

Multidimensional Case

Assume you have two multidimensional gaussian distributions.

Sum of Gaussians

$\mu ={\mu }_1+{\mu }_2$ $\Sigma ={\Sigma }_1+{\Sigma }_2$

• Where ${\Sigma }_1$ is the Covariance Matrix for the first distribution, and ${\Sigma }_2$ for the second distribution

Product of Gaussians

$\mu ={\Sigma }_2({\Sigma }_1+{\Sigma }_2{)}^{-1}{\mu }_1+{\Sigma }_1({\Sigma }_1+{\Sigma }_2{)}^{-1}{\mu }_2$ $\Sigma ={\Sigma }_1({\Sigma }_1+{\Sigma }_2{)}^{-1}{\Sigma }_2$

• Note here that ${\Sigma }_1$ is the Covariance Matrix of the first multidimensional gaussian distribution (we aren’t indexing into it, it isn’t a scalar), and ${\Sigma }_2$ likewise for the second multidimensional gaussian distribution.

Example

Consider a random variable $x\sim N({\mu }_x,{\Sigma }_{xx})$ , and $y=Ax+b+w$, where $A,b$ are the linear coefficients and bias, $w\sim N(0,R)$.

• where $N$ is the Normal Distribution

Then, $y$’s distribution is given by $p(y)=N(A{\mu }_x+b,R+A{\Sigma }_{xx}{A}^T)$

Related

• Kalman Filter