APPENDIX

APPENDIX

Excel

Model Density function and parameters
Weibull Density function
lowercase f (lowercase x) = lowercase sigma divided by lowercase alpha (lowercase x minus lowercase mu divided by lowercase alpha) superscript {lowercase gamma minus 1} exp (negative ((lowercase x minus lowercase mu) divided by lowercase alpha) superscript {lowercase gamma}) for lowercase x greater than or equal to lowercase mu; lowercase gamma, lowercase alpha greater than 0
where
g is the shape parameter
μ is the location parameter
α is the scale parameter
BetaGeneral
(Beta Generalized)
Density function
lowercase f (lowercase x) = ((lowercase x minus min) superscript {lowercase alpha subscript {2} minus 1} (max minus lowercase x) superscript {lowercase alpha subscript {2} minus 1}) divided by (uppercase b (lowercase alpha subscript {1}, lowercase alpha subscript {2}) (max minus min) superscript {lowercase alpha subscript {1} plus lowercase alpha subscript {2} minus 1})
where
α1 is the continuous shape parameter α1 > 0
α2 is the continuous shape parameter α2 > 0
min is the continuous boundary parameter min < max
max is the continuous boundary parameter
Gamma Density function
lowercase f (lowercase x) = 1 divided by (lowercase beta uppercase gamma (lowercase alpha)) (lowercase x divided by lowercase beta) superscript {lowercase alpha minus 1} lowercase e superscript {negative lowercase x over lowercase beta}
with gamma function uppercase gamma (lowercase alpha) = integration from 0 to infinity of lowercase u superscript {lowercase alpha minus 1} lowercase e superscript {negative lowercase u} du
where
α is the continuous shape parameter α > 0
β is the continuous shape parameter β > 0
Inverse Gaussian Density function
lowercase f (lowercase x) = square root of (lowercase lambda divided by (2 lowercase pi lowercase x superscript {3})lowercase e) superscript {negative [(lowercase lambda (lowercase x minus lowercase mu) superscript {2}) divided by (2 lowercase mu superscript {2} lowercase x)]}
where both μ and λ are positive continuous parameters
Log normal Density function
lowercase f (lowercase x) = 1 divided by (lowercase x square root of (2 lowercase pi) lowercase sigma prime) lowercase e superscript {negative 1 divided by 2 [(ln lowercase x minus lowercase mu prime) divided by lowercase sigma prime] superscript {2}}
with lowercase mu prime = ln [lowercase mu superscript {2} divided by square root of (lowercase sigma superscript {2} plus lowercase mu superscript {2})] and lowercase sigma prime = square root of (lon [1 plus (lowercase sigma over lowercase mu) superscript {2}])
where both μ and σ are positive continuous parameters