Table 1.

Definitions of alternative discrete distributions

Distribution name	Probability distribution function
Exponential	$(1-{\text{e}}^{-\mathit{\lambda }}){\text{e}}^{\mathit{\lambda }{x}_0}{\text{e}}^{-\mathit{\lambda }x}$
Stretched exponential (Weibull)	$\frac{1}{{\sum }_{x_0}^{\infty }(q^{x^{\mathit{\beta }}}-q^{{(x+1)}^{\mathit{\beta }}})}{q}^{x^{\mathit{\beta }}}-q^{{(x+1)}^{\mathit{\beta }}}$
Log-normal	$\sqrt{\frac{2}{\mathit{\pi }{\mathit{\sigma }}^2}}{\left[\text{erfc}(\frac{\text{ln}{x}_0-\mathit{\mu }}{\sqrt{2}\mathit{\sigma }})\right]}^{-1}\frac{1}{x}exp\left[-\frac{{(\text{ln}x-\mathit{\mu })}^2}{2{\mathit{\sigma }}^2}\right]$
Tsallis	$\frac{1}{{\sum }_{x_0}^{\infty }{(1+x/\mathit{\sigma })}^{-\mathit{\theta }-1}}{(1+x/\mathit{\sigma })}^{-\mathit{\theta }-1}$
Yule	$(\mathit{\alpha }-1)\frac{\mathrm{\Gamma }(x_0+\mathit{\alpha }-1)}{\mathrm{\Gamma }(x_0)}\frac{\mathrm{\Gamma }(x)}{\mathrm{\Gamma }(x+\mathit{\alpha })}$
Digamma	$\frac{{\text{e}}^{-\mathit{\mu }{k}_0\mathit{\psi }(x+k_0)}}{{\sum }_{x_0}^{\infty }{\text{e}}^{-\mathit{\mu }{k}_0\mathit{\psi }(x+k_0)}}$
Power law with exponential cut-off	${({\sum }_{x_0}^{\infty }{x}^{-\mathit{\alpha }}{\text{e}}^{-\mathit{\lambda }x})}^{-1}{x}^{-\mathit{\alpha }}{\text{e}}^{-\mathit{\lambda }x}$

The distributions have been normalized to ensure that the total probability in the domain $[x_0,+\infty ]$ is 1. Discrete log-normal distribution is approximated by rounding the continuous log-normally distributed reals to the nearest integers. For Tsallis distribution, we use a parametrization considered by Shalizi (2007)