User:Karoolc/sandbox

Definition

The split normal distribution arises from merging two opposite halves of two probability density functions (PDFs) of normal distributions in their common mode.

The PDF of the split normal distribution is given by^[1]

${\displaystyle f(x;\mu ,\sigma _{1},\sigma _{2})=A\exp(-{\frac {(x-\mu )^{2}}{2\sigma _{1}^{2}}})\quad {\text{if }}x<\mu }$

${\displaystyle f(x;\mu ,\sigma _{1},\sigma _{2})=A\exp(-{\frac {(x-\mu )^{2}}{2\sigma _{2}^{2}}})\quad {\text{otherwise}}}$

where

${\displaystyle \quad A=2/\pi (\sigma _{1}+\sigma _{2})^{-1}}$.

Discusssion

The split normal distribution results from merging two halves of normal distributions. In a general case the 'parent' normal distributions can have different variances which implies that the joined PDF would not be continuous. To insure that the resulting PDF is continous, the normalizing constant A is used. The constant also insures that the PDF integrates to 1.

In a special case when ${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{*}^{2}}$ the split normal distribution reduces to normal distribution with variance ${\displaystyle \sigma _{*}^{2}}$.

When σ₂≠σ₁ the constant A it is different from the constant of normal distribution. However, when ${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{*}^{2}}$ the constants are equal.

The sign of its third central moment is determined by the difference (σ₂-σ₁). If this difference is positive, the distribution is skewed to the right and if negative, then it is skewed to the left.

Other properties of the split normal density were discussed by Johnson et al.^[1] and Julio^[2].