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
, , , σ > 0
E(X) = μ
Var(X) = σ2
Beta
, , α > 0, β > 0
E(X) =
Var(X) =
where is the Beta function.
Exponential
, x > 0, σ > 0
E(X) = σ
Var(X) = σ2
Gamma
, x > 0, α > 0, σ > 0
E(X) = ασ
Var(X) = ασ2
where is the gamma function.
Johnson
Johnson Su
 
, , θ > 0, δ > 0
Johnson Sb
 
, θ < x < θ+σ, σ > 0
Johnson Sl
 
, for   x > θ if σ = 1, x < θ if σ = -1
where is the standard normal probability density function.
Lognormal
, x > 0, , σ > 0
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
, α > 0, β > 0
E(X) =
Var(X) =
where is the gamma function.

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Help created on 7/12/2018