The probability density function of a normal distribution (Gaussian distribution), N(μ, σ), with mean μ and standard deviation σ > 0, for -∞ < x < ∞, is given below:

μ= σ=

pdf of the Normal, N_μ,σ

Probability density function:

f(x) =	1	-(x-μ)2/2.σ2  e
	√(2.π) σ

 

and of course

_-∞∫^+∞ f(x) dx = 1

MML

ε: (AoM)
μ: to uniform
σ: to h σ ~ 1/σ
-0.75 -2.04 -1.04 0.0 1.45 0.32 -0.44 1.24 -0.97 1.17

Notes

All text books on probability and statistics will cover the basic properties of the normal distribution, e.g.,

• P. L. Meyer. Introductory Probability and Statistical Application, Addison Wesley, 1970.

The study of the normal distribution (and the multi-state distribution) in the context of (unsupervised) classification — also known as clustering, numerical taxonomy, and mixture modelling — by Wallace and Boulton is one of the first applications of minimum message length (MML) encoding to a practical machine-learning problem yielding a useful computer program, Snob:

• C. S. Wallace & D. M. Boulton. An Information Measure for Classification, The Computer Journal, 11(2), pp.185-194, August 1968.

• Also see the Special Issue on Clustering and Classification, The Computer Journal, F. Murtagh (ed), 41(8), 1998.

  Contains several papers on classification, marks the 30^th anniversary of the Wallace & Boulton (1968) paper, and also includes a new paper by Wallace on modelling spatially correlated data.