Correction: A Novel Generalized Normal Distribution for Human Longevity and other Negatively Skewed Data

In Table 1, the “Formula” column for “CDF” is incorrect. Please see the corrected Table 1 here.

Table 1. Properties of the distribution.

Quantity	Formula
Parameters	0 < σ < μ < λ
Domain	x є (0,λ)
PDF	$\text{f}(\text{x})=\frac{(1-\mu /\mathrm{\lambda })}{\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)\sqrt{2\mathrm{\pi }}\mathrm{\sigma }{(1-\text{x}/\mathrm{\lambda })}^2}\text{exp}\left\{-\frac{{(\text{x}-\mathrm{\mu })}^2}{{2\mathrm{\sigma }}^2{(1-\text{x}/\mathrm{\lambda })}^2}\right\}$
CDF	$\text{F}(\text{x})=1-\frac{1-\mathrm{\Phi }\left(\frac{\text{x}-\mathrm{\mu }}{\mathrm{\sigma }(1-\text{x}/\mathrm{\lambda })}\right)}{\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)}$
Hazard	$\frac{(1-\mathrm{\mu }/\mathrm{\lambda })}{\sigma {(1-\text{x}/\mathrm{\lambda })}^2}\cdot \frac{\varphi \left[\frac{\text{x}-\mathrm{\mu }}{\mathrm{\sigma }(1-\text{x}/\mathrm{\lambda })}\right]}{1-\mathrm{\Phi }\left[\frac{\text{x}-\mathrm{\mu }}{\mathrm{\sigma }(1-\text{x}/\mathrm{\lambda })}\right]}$
Quantile	$\mathrm{\lambda }\cdot \frac{\mathrm{\mu }-{\mathrm{\sigma }\mathrm{\Phi }}^{-1}\left[\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)(1-\text{p})\right]}{\mathrm{\lambda }-{\mathrm{\sigma }\mathrm{\Phi }}^{-1}\left[\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)(1-\text{p})\right]}$
Median	$\frac{\mu -\sigma {\mathrm{\Phi }}^{-1}\left[\frac{1}{2}\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)\right]}{1-\frac{\sigma }{\lambda }{\mathrm{\Phi }}^{-1}\left[\frac{1}{2}\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)\right]}\approx \mu$
Mode	$\mathrm{\lambda }\left\{1+\frac{(\mathrm{\mu }-\mathrm{\lambda })(-\mathrm{\lambda }+\sqrt{{\mathrm{\lambda }}^2+8{\mathrm{\sigma }}^2})}{4{\mathrm{\sigma }}^2}\right\}$
Mean	$\underset{0}{\overset{\lambda }{\int }}\frac{\text{x}(1-\mathrm{\mu }/\mathrm{\lambda })}{\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)\sqrt{2\mathrm{\pi }}\mathrm{\sigma }(1-\text{x}/\mathrm{\lambda }{)}^2}\text{exp}\left\{-\frac{{(\text{x}-\mathrm{\mu })}^2}{{2\mathrm{\sigma }}^2{(1-\text{x}/\mathrm{\lambda })}^2}\right\}\text{dx}$
Variance	$\underset{0}{\overset{\lambda }{\int }}\frac{{(\text{x}-\text{E}(\text{X}))}^2(1-\mathrm{\mu }/\mathrm{\lambda })}{\mathrm{\Phi }\left(\frac{\mathrm{\mu }}{\mathrm{\sigma }}\right)\sqrt{2\mathrm{\pi }}\mathrm{\sigma }(1-\text{x}/\mathrm{\lambda }{)}^2}\text{exp}\left\{-\frac{{(\text{x}-\mathrm{\mu })}^2}{{2\mathrm{\sigma }}^2{(1-\text{x}/\mathrm{\lambda })}^2}\right\}\text{dx}$