Log-Cauchy distribution

Log-Cauchy
	Probability density function
	Cumulative distribution function
Parameters	(real); (real)
Support
PDF
CDF
Mean	infinite
Median
Variance	infinite
Skewness	does not exist
Excess kurtosis	does not exist
MGF	does not exist

In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution.^[1]

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[1]

Log-Cauchy
Probability density function
Cumulative distribution function
Parameters	$\mu$ (real) $\displaystyle \sigma >0\!$ (real)
Support	$\displaystyle x\in (0,+\infty )\!$
PDF	${1 \over x\pi }\left[{\sigma \over (\ln x-\mu )^{2}+\sigma ^{2}}\right],\ \ x>0$
CDF	${\frac {1}{\pi }}\arctan \left({\frac {\ln x-\mu }{\sigma }}\right)+{\frac {1}{2}},\ \ x>0$
Mean	infinite
Median	$e^{\mu }\,$
Variance	infinite
Skewness	does not exist
Excess kurtosis	does not exist
MGF	does not exist

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Log-Cauchy distribution

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