Normal (Gaussian) Distribution

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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, Nμ,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  

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