# Derivation of GMM

## Derivation of one single-variable gaussian distribution

Considering a generative model, let $P$ denote the generated probability distribution, ${x}_{i=1}^n$ denote the data set, and $\mu$ and $\sigma$ denotes the parameters of a generated Gaussian Distribution. We will go over a max-likelihood process to find the parameters of the generated distribution.

**Independence Assumption:**
$x_i\perp x_j\ \forall i,j \ \mbox{s.j} \ 1\le i,j\le n$

## Result of one multi-variable gaussian distribution

**Multi-variate Gaussian Distribution Probability Function**

## Result of EM algorithm

**Gaussian Mixture**

**Latent Variable**

**Result**

## Ref

- Coursera Robotics Week 1 Lectures