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Power Lognormal Distribution

NIST/SEMATECH Section 1.3.6.6.14 Power Lognormal Distribution

PDF

PowLN(p=1, s=1) PDF 2 4 6 8 10 12 14 16 18 20 x 0 0.2 0.4 0.6 f(x)

CDF

PowLN(p=1, s=1) CDF 2 4 6 8 10 12 14 16 18 20 x 0 0.2 0.4 0.6 0.8 1 F(x)

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PDF

CDF

Formulas

Probability Density Function

f(x)=pxσ ϕ ⁣(ln⁡xσ)[Φ ⁣(−ln⁡xσ)]p−1,x>0f(x) = \frac{p}{x\sigma}\,\phi\!\left(\frac{\ln x}{\sigma}\right)\left[\Phi\!\left(-\frac{\ln x}{\sigma}\right)\right]^{p-1}, \quad x > 0

Cumulative Distribution Function

F(x)=1−[Φ ⁣(−ln⁡xσ)]pF(x) = 1 - \left[\Phi\!\left(-\frac{\ln x}{\sigma}\right)\right]^p

Properties

Mean

no closed form\text{no closed form}

Variance

no closed form\text{no closed form}

Overview

The power lognormal distribution generalizes the lognormal distribution for reliability analysis, modeling the minimum of pp independent lognormal lifetimes with scale σ\sigma. It provides additional shape flexibility beyond the lognormal.

Related Distributions

Lognormal Distribution Power Normal Distribution Weibull Distribution