Gaussian and Truncated Gaussian

Everybody knows about Gaussian distribution, and Gaussian is very popular in Bayesian world and even in our life. This article summaries typical operation of Gaussian, and something about Truncated Guassian distribution.

pdf(probability density function) and cdf(cumulative density function) of Gaussian distribution

Sum (or substraction) of two independent Gaussian random variables

Please take care upper formula only works when x1 and x2 are independent. And it’s easy to get the distribution for variable x=x1-x2 See [here] for the detils of inference

Product of two Gaussian pdf

Please take care x is no longer a gaussian distribution. And you can find it’s very elegant to use ‘precision’ and ‘precision adjusted mean’ for Gaussian operation like multiply and division. See [here] for the detils of inference

division of two Gaussian pdf

And it’s common to calculate the intergral of the product of two gaussian distribution

Truncated Gaussian

Truncated Gaussian distribution is very simple: it’s just one conditional (Gaussian) distribution. Suppose variable x belongs to Gaussian distribution, then x conditional on x belongs to (a, b) has a truncated Gaussian distribution.