Overview
The normal distribution is the standard probability distribution known as the Gaussian, or the bell curve. It is one or the most common distributions used in statistical modeling, usually because of its use in the central limit theorem. The normal distribution exhibits a characteristic bell shape.Formal Definition
The normal density function is given by
{% f(x) = \sqrt{1/2\pi \sigma^2 } \times e ^{-0.5 [(x-\mu)/\sigma]^2} %}
For multivariable distributions
{% f(\vec{x}| \vec{\mu}, \Sigma) = \frac{1}{(2\pi )^{D/2} | \Sigma | ^{1/2}} exp[-\frac{1}{2} (\vec{x} - \vec{\mu})^T \Sigma ^{-1} (\vec{x} - \vec{\mu}) ] %}
(see Murphy chpt 4)
Topics
- Maximum Entropy - the normal distribution is the distribution that results from maximizing the entropy of a distribution with mean 0 and variance 1.
- Sum of Normals
- Central Limit Theorem
- Linear Algebra Formulations
- Normal Vector Space