The Normal Distribution is the most important (but by no means the only) distribution for continuous values. Probability density:
2 1 1 |x - mu| f(x) = ---------------- exp( - - |------| ) sqrt(2 pi) sigma 2 |sigma |
- log_{e} f(x) :
2 1 |x - mu| -ln f(x) = ln(sqrt(2 pi) sigma) + - |------| 2 |sigma |
This document is online at http://www.csse.monash.edu.au/~lloyd/Archive/2005-05-Normal/index.shtml and contains hyper-links to other resources.
mu_{ML} | = | (x_{1} + x_{2} + . . . + x_{n}) / n |
v_{ML} | = | SUM_{i=1..n} (x_{i} - mu_{ML})^{2}/n |
and sigma = sqrt(v).
Consider n=1, and n=2.
with prior h(mu,v) ~ 1/v
mu_{MML} | = | (x_{1} + x_{2} + . . . + x_{n}) / n | = | mu_{ML} |
v_{MML} | = | SUM_{i=1..n} (x_{i} - mu)^{2}/(n-1) |
and sigma = sqrt(v).
Consider n=1, and n=2.