Table A1.

The PDF and CDF of five kinds of distributions.

Distribution	PDF/CDF	Parameters
Weibull	$f(x;a,b)=\frac{b}{a}{(\frac{x}{a})}^{b-1}\exp [-{(\frac{x}{a})}^b],x\ge 0$	a > 0 scale parameter b > 0 shape parameter
	$F(x;a,b)=1-\exp [-{(\frac{x}{a})}^b]$
Gamma	$f(x;a,b)=x^{a-1}\exp (-\frac{x}{b})/b^a\Gamma (a),x>0$	a > 0 shape parameter b > 0 scale parameter
	$F(x;a,b)=\frac{1}{b^a\Gamma (a)}{\displaystyle {\int }_0^x{t}^{a-1}\exp (-\frac{x}{b})}dt,x>0$
Lognormal	$f(x;a,b)=\frac{1}{xa\sqrt{2\pi }}exp(-\frac{{(Inx-b)}^2}{2a^2}),x>0$	a > 0 scale parameter b > 0 location parameter
	$F(x;a,b)=\frac{1}{a\sqrt{2\pi }}{\displaystyle {\int }_0^x\frac{1}{t}}\exp (-\frac{{(Int-b)}^2}{2a^2})dt$
Log-logistic	$f(x;a,b)=\frac{(b/a){(x/a)}^{b-1}}{{(1+{(x/a)}^b)}^2},x>0$	a > 0 scale parameter b > 0 shape parameter
	$F(x;a,b)=\frac{x^b}{a^b+x^b}$
Inverse Gaussian	$f(x;a,b)={\left[\frac{b}{2\pi x^3}\right]}^{1/2}\exp \left[\frac{-b{(x-a)}^2}{2a^{2}x}\right],x>0$	a > 0 scale parameter b > 0 shape parameter
	$F(x;a,b)=\Phi \left[\sqrt{\frac{b}{x}}(\frac{x}{a}-1)\right]+\left[\exp (\frac{2b}{a})\right]\Phi \left[-\sqrt{\frac{b}{x}}(\frac{x}{a}+1)\right]$