Deriving Gaussian Distribution

Gaussian is very important distribution. During this post, we will discuss the detail of Gaussian distribution by deriving it, calculate the integral value and do MLE (Maximum Likelihood Estimation). To derive Gaussian distribution, it is more difficult if we do it in cartesian coordinate. Thus, we will use polar coordinate. Before we derive the Gaussian using polar coordinate, let’s talk about how to change the coordinate system from cartesian to polar coordinate system first.

(1) Changing coordinate system from cartesian to polar coordinate

Changing coordinate system from catersian to polar coordinate is useful, such as when we calculate integral of certain function, in certain case, we prefer to use polar coordinate system because it will be away easier to calculate. To do that, we can use Jacobian matrix. Jacobian matrix actually defines partial derivative of a vector with respect to another vector. In our case changing cartesian coordinate to polar coordinate, the Jacobian matrix of (x,y) in cartesian coordinate with respect to (r,\theta) in polar coordinate is:

J = \begin{bmatrix} \frac{\delta x}{\delta r} \,\frac{\delta x}{\delta \theta} \\\frac{\delta y}{\delta r} \,\frac{\delta y}{\delta \theta}\end{bmatrix} Continue reading “Deriving Gaussian Distribution”