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Two-Piece Normal Distribution.pdf

1405.4995.pdf
Preview of Two-Piece Normal Distribution
🔗 Source: arxiv.org
📊 Size: 176 KB
👤 Author: Kenneth F. Wallis
⬇️ Downloads: 424

Summary

The two-piece normal distribution originated in Gustav Theodor Fechner's 1897 book "Kollektivmasslehre". Fechner introduced the "Zweispaltiges" or "Zweiseitige Gauss'sche Gesetz", which translates to the "two-sided Gaussian law". The distribution is formed by taking the left half of a normal distribution with parameters (µ,σ1) and the right half of a normal distribution with parameters (µ,σ2), and scaling them to give a common value at the mode, µ.

The probability density function (PDF) of the two-piece normal distribution is given by:
f(x) =
{ Aexp[−(x −µ)2/2σ2 1], x ≤µ
{ Aexp[−(x −µ)2/2σ2 2], x ≥µ

where A = (√2π(σ1 + σ2)/2)−1. The distribution has parameters µ, σ1, and σ2, and is positively skewed when σ1 < σ2. The mean and variance of the distribution are given by:
E(X) = µ + √2/π(σ2 −σ1)
var(X) = (1 − 2/π)(σ2 −σ1)2 + σ1σ2

The distribution was rediscovered by Francis Ysidro Edgeworth in 1899 and later discussed by Karl Pearson in 1905, who denied Fechner's originality. However, this denial is shown to be without foundation. The distribution has since been independently rediscovered several times, with extensions and applications in various fields.

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Description

The two-piece normal distribution originated in Fechner's 1897 work. It was later rediscovered and generalized. Fechner's originality was denied by Karl Pearson.

Technical Information

  • File Format: PDF
  • File Size: 176 KB
  • Pages: 8
  • Language: EN
  • Author: Kenneth F. Wallis
  • Total Downloads: 424
  • Last Updated: 3 hours ago

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