FATIGUE LIFE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.FatigueLife({"gamma": *, "loc": *, "scale": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{FatigueLife}(\gamma ,\text{Loc},\text{Sc})$$

Distribution Domain

$$x\in (\text{Loc},\mathrm{\infty })$$

Parameters Domain and Constraints

$$\gamma \in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R},\text{Sc}\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\mathrm{\Phi }\left(\frac{\sqrt{z(x)}-\sqrt{\frac{1}{z(x)}}}{\gamma }\right)$$

Probability Density Function

$$f_X\left(x\right)=\frac{\sqrt{z(x)}+\sqrt{\frac{1}{z(x)}}}{2\gamma z(x)}\varphi \left(\frac{\sqrt{z(x)}-\sqrt{\frac{1}{z(x)}}}{\gamma }\right)$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\text{Sc}\frac{1}{4}{[\gamma {\mathrm{\Phi }}^{-1}(u)+\sqrt{4+{(\gamma {\mathrm{\Phi }}^{-1}(u))}^2}]}^2$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot {\tilde{\mu }}_1^{\prime }=\text{Loc}+\text{Sc}(1+\frac{{\gamma }^2}{2})$$

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2\cdot ({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})={\text{Sc}}^2{\gamma }^2(1+\frac{5{\gamma }^2}{4})$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{4\gamma (6+11{\gamma }^2)}{(4+5{\gamma }^2{)}^{1.5}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=3+\frac{6{\gamma }^2(93{\gamma }^2+40)}{(5{\gamma }^2+4{)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{1}{4}{[\gamma {\mathrm{\Phi }}^{-1}\left(1/2\right)+\sqrt{4+{\left(\gamma {\mathrm{\Phi }}^{-1}\left(1/2\right)\right)}^2}]}^2$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• $\varphi \left(x\right):\text{PDF normal standard distribution}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document