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Split normal distribution

In probability theory and statistics, the split normal distribution also known as the two-piece normal distribution results from joining at the mode the corresponding halves of two normal distributions with the same mode but different variances. It is claimed by Johnson et al. that this distribution was introduced by Gibbons and Mylroie and by John. But…

Applications, Definition & Estimation of parameters

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Applications

Fan chartInflationInflation targeting
3 related topics

Definition

Probability density functionsNormal distributionContinuousIntegratesNormalizing constant
5 related topics

Estimation of parameters

Maximum likelihoodBayesian estimation
2 related topics

Alternative formulations

Bank of England's
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Mean
μ + 2 / π ( σ 2 − σ 1 ) {\displaystyle \mu +{\sqrt {2/\pi }}(\sigma _{2}-\sigma _{1})}
Mode
μ {\displaystyle \mu }
Notation
S N ( μ , σ 1 , σ 2 ) {\displaystyle {\mathcal {SN}}(\mu ,\,\sigma _{1},\sigma _{2})}
Parameters
μ ∈ ℜ {\displaystyle \mu \in \Re } — mode (location, real) σ 1 > 0 {\displaystyle \sigma _{1}>0} — left-hand-side standard deviation (scale, real) σ 2 > 0 {\displaystyle \sigma…
PDF
A exp ⁡ ( − ( x − μ ) 2 2 σ 1 2 ) if x < μ {\displaystyle A\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma _{1}^{2}}}\right)\quad {\text{if }}x<\mu } A exp ⁡ ( − ( x − μ ) 2 2 σ 2 2 )…
Skewness
γ 3 = 2 π ( σ 2 − σ 1 ) [ ( 4 π − 1 ) ( σ 2 − σ 1 ) 2 + σ 1 σ 2 ] {\displaystyle \gamma _{3}={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\ri…

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Overview

Definition

Alternative formulations

Multivariate Extensions

Estimation of parameters

Applications

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Map overview Semantic statistics

Split normal distribution

Nodes25
Edges24
Triples20
Avg. degree1.92
Density0.08
Components1

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Split normal distribution

Top relations

related to Discussion · 4
Split normal distribution → In, PDF, The, To
related to Multivariate Extensions · 4
Split normal distribution → Larsson, The, They, Villani
related to Definition · 3
Split normal distribution → PDFs, The, The PDF
Mean · 1
Split normal distribution → μ + 2 / π ( σ 2 − σ 1 ) {\displaystyle \mu +{\sqrt {2/\pi }}(\sigma _{2}-\sigma _{1})}
Mode · 1
Split normal distribution → μ {\displaystyle \mu }
Notation · 1
Split normal distribution → S N ( μ , σ 1 , σ 2 ) {\displaystyle {\mathcal {SN}}(\mu ,\,\sigma _{1},\sigma _{2})}
Parameters · 1
Split normal distribution → μ ∈ ℜ {\displaystyle \mu \in \Re } — mode (location, real) σ 1 0 {\displaystyle \sigma _{1}0} — left-hand-side standard deviation (scale, real) σ 2 0 {\displaystyle \sigma…
PDF · 1
Split normal distribution → A exp ⁡ ( − ( x − μ ) 2 2 σ 1 2 ) if x μ {\displaystyle A\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma _{1}^{2}}}\right)\quad {\text{if }}x\mu } A exp ⁡ ( − ( x − μ ) 2 2 σ 2 2 )…
Skewness · 1
Split normal distribution → γ 3 = 2 π ( σ 2 − σ 1 ) [ ( 4 π − 1 ) ( σ 2 − σ 1 ) 2 + σ 1 σ 2 ] {\displaystyle \gamma _{3}={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\ri…
Support · 1
Split normal distribution → x ∈ ℜ {\displaystyle x\in \Re }

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Important terminology

distribution mode normal split parameters parameter two displaystyle sigma likelihood distributions different scale mu pdf skewness formulation dispersion results halves

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Split normal distributionMeanμ + 2 / π ( σ 2 − σ 1 ) {\displaystyle \mu +{\sqrt {2/\pi }}(\sigma _{2}-\sigma _{1})}1.00infobox
Split normal distributionModeμ {\displaystyle \mu }1.00infobox
Split normal distributionNotationS N ( μ , σ 1 , σ 2 ) {\displaystyle {\mathcal {SN}}(\mu ,\,\sigma _{1},\sigma _{2})}1.00infobox
Split normal distributionParametersμ ∈ ℜ {\displaystyle \mu \in \Re } — mode (location, real) σ 1 > 0 {\displaystyle \sigma _{1}>0} — left-hand-side standard deviation (scale, real) σ 2 > 0 {\displaystyle \sigma…1.00infobox
Split normal distributionPDFA exp ⁡ ( − ( x − μ ) 2 2 σ 1 2 ) if x < μ {\displaystyle A\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma _{1}^{2}}}\right)\quad {\text{if }}x<\mu } A exp ⁡ ( − ( x − μ ) 2 2 σ 2 2 )…1.00infobox
Split normal distributionSkewnessγ 3 = 2 π ( σ 2 − σ 1 ) [ ( 4 π − 1 ) ( σ 2 − σ 1 ) 2 + σ 1 σ 2 ] {\displaystyle \gamma _{3}={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\ri…1.00infobox
Split normal distributionSupportx ∈ ℜ {\displaystyle x\in \Re }1.00infobox
Split normal distributionVariance( 1 − 2 / π ) ( σ 2 − σ 1 ) 2 + σ 1 σ 2 {\displaystyle (1-2/\pi )(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}}1.00infobox
Split normal distributionhas applicationThe0.60section
Split normal distributionrelated to DefinitionThe0.60section
Split normal distributionrelated to DefinitionPDFs0.60section
Split normal distributionrelated to DefinitionThe PDF0.60section

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