Parameterizations for Distributions

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Quality and Process Methods • Process Capability • Statistical Details for the Process Capability Platform • Parameterizations for Distributions

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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 distributions.

Normal

E(X) = μ

Var(X) = σ2

Beta

E(X) =

Var(X) =

where

is the Beta function.

Exponential

E(X) = σ

Var(X) = σ2

Gamma

E(X) = ασ

Var(X) = ασ2

where

is the gamma function.

Johnson

Johnson Su

Johnson Sb

Johnson Sl

where

is the standard normal probability density function.

Lognormal

E(X) =

Var(X) =

Mixture of Normals

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

E(X) =

Var(X) =

where μi, σi, and πi are the respective mean, standard deviation, and proportion for the ith group, and

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.

Weibull

E(X) =

Var(X) =

where

is the gamma function.

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Help created on 3/19/2020