News Contents: Benini distribution

2011-04-07

Benini
Probability density function No image available
Cumulative distribution function No image available
parameters:	$α > 0$ shape (real) $β > 0$ shape (real) $σ > 0$ scale (real)
support:	$x > σ$
pdf:	$e^{-\alpha\log{\frac{x}{\sigma}}-\beta\log\left[{\frac{x}{\sigma}}\right]^2} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)$
cdf:	$1-e^{-\alpha\log{\frac{x}{\sigma}}-\beta\log[{\frac{x}{\sigma}}]^2}$
mean:	$\sigma+\tfrac{\sigma}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta}}\right)$ where $H n (x)$ is the "probabilists' Hermite polynomials"
median:	$\sigma \left(e^{\frac{-\alpha+\sqrt{\alpha^2+\beta\log{16}}}{2\beta}}\right)$
mode:
variance:	$\left(\sigma^2+\tfrac{2\sigma^2}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-2+\alpha}{\sqrt{2\beta}}\right)\right)-\mu^2$
skewness:
ex.kurtosis:
entropy:
mgf:
cf:

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