Publication date: 03/23/2021

Parameterizations for Distributions

This section gives the density functions f for the distributions used in the Process Capability platform. It also gives expected values and variances for all but the Johnson and SHASH distributions.

Normal

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E(X) = μ

Var(X) = σ2

Beta

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E(X) = Equation shown here

Var(X) = Equation shown here

where B(·) is the Beta function.

Exponential

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E(X) = σ

Var(X) = σ2

Gamma

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E(X) = ασ

Var(X) = ασ2

where Γ(·) is the gamma function.

Johnson

Johnson Su

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Johnson Sb

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Johnson Sl

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where φ(·) is the standard normal probability density function.

Lognormal

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E(X) =Equation shown here

Var(X) =Equation shown here

Mixture of Normals

The Mixture of 2 Normals and Mixture of 3 Normals options for Distribution share the following parameterization:

Equation shown here

E(X) =Equation shown here

Var(X) =Equation shown here

where μi, σi, and πi are the respective mean, standard deviation, and proportion for the ith group, and φ(·) is the standard normal probability density function. For the Mixture of 2 Normals, k is equal to 2. For the Mixture of 3 Normals distribution, k is equal to 3. A separate mean, standard deviation, and proportion of the whole is estimated for each group in the mixture.

SHASH

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where

φ(·) is the standard normal pdf

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Note: When γ = 0 and δ = 1, the SHASH distribution is equivalent to the normal distribution with location θ and scale σ.

Weibull

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E(X) =Equation shown here

Var(X) =Equation shown here

where Γ(·) is the gamma function.

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