# Pattern Recognition and Machine Learning (Information Science and Statistics) - 2006

## Details

Title : Pattern Recognition and Machine Learning (Information Science and Statistics) Author(s): Bishop, Christopher M. Link(s) :

## Rough Notes

### Probability Distributions

#### 2.3. The Gaussian Distribution

- The Central Limit Theorem (CLT) states that under mild conditions, the sum of a set of random variables tends to a Gaussian as the terms tend to infinity.
- The multivariate Gaussian, characterized by a mean vector \(\mu\) and covariance matrix \(\Sigma\), induces the probability density \(p(x\:;\:\mu, \Sigma)\) where they relate to \(x\) in a quadratic form \((x-\mu)^T\Sigma^{-1}(x-\mu)\), which is also the square of the Mahalanobis distance from \(\mu\) to \(x\).
- All eigenvalues of \(\Sigma\) must be positive for the density to be properly normalized.
- Assuming the transformation \(y = U(x-\mu)\) where \(U\) is a matrix whose columns are the eigenvectors of \(\Sigma\), we have that \(y\) is Gaussian with independent components with the variance for coordinate \(y_i\) being the corresponding eigenvalue \(\lambda_i\). Using the independence of the coordinates, and the normalization constant of 1D Gaussians.
- [#TODO Complete from 2.3.1]