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.
Characterization The log-Cauchy distribution is a special case of the log-t distribution where the degrees of freedom parameter is equal to 1.
Probability density function The log-Cauchy distribution has the probability density function:
f ( x ; μ , σ ) = 1 x π σ [ 1 + ( ln x − μ σ ) 2 ] , x > 0 = 1 x π [ σ ( ln x − μ ) 2 + σ 2 ] , x > 0 {\displaystyle {\begin{aligned}f(x;\mu ,\sigma )&={\frac {1}{x\pi \sigma \left[1+\left({\frac {\ln x-\mu }{\sigma }}\right)^{2}\right]}},\ \ x>0\\&={1 \over x\pi }\left[{\sigma \over (\ln x-\mu )^{2}+\sigma ^{2}}\right],\ \ x>0\end{aligned}}}
where μ {\displaystyle \mu } is a real number and σ > 0 {\displaystyle \sigma >0} . If σ {\displaystyle \sigma } is known, the scale parameter is e μ {\displaystyle e^{\mu }} . μ {\displaystyle \mu } and σ {\displaystyle \sigma } correspond to the location parameter and scale parameter of the associated Cauchy distribution. Some authors define μ {\displaystyle \mu } and σ {\displaystyle \sigma } as the location and scale parameters, respectively, of the log-Cauchy distribution. For μ = 0 {\displaystyle \mu =0} and σ = 1 {\displaystyle \sigma =1} , corresponding to a standard Cauchy distribution, the probability density function reduces to:
f ( x ; 0 , 1 ) = 1 x π [ 1 + ( ln x ) 2 ] , x > 0 {\displaystyle f(x;0,1)={\frac {1}{x\pi [1+(\ln x)^{2}]}},\ \ x>0}
Cumulative distribution function The cumulative distribution function (cdf) when μ = 0 {\displaystyle \mu =0} and σ = 1 {\displaystyle \sigma =1} is:
F ( x ; 0 , 1 ) = 1 2 + 1 π arctan ( ln x ) , x > 0 {\displaystyle F(x;0,1)={\frac {1}{2}}+{\frac {1}{\pi }}\arctan(\ln x),\ \ x>0}
Survival function The survival function when μ = 0 {\displaystyle \mu =0} and σ = 1 {\displaystyle \sigma =1} is:
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