Template:Loglogistic probability density function

Loglogistic Probability Density Function

The loglogistic distribution is a two-parameter distribution with parameters [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \sigma }[/math] . The [math]\displaystyle{ pdf }[/math] for this distribution is given by:

[math]\displaystyle{ f(t)=\frac{{{e}^{z}}}{\sigma {t}{{(1+{{e}^{z}})}^{2}}} }[/math]

where:

[math]\displaystyle{ z=\frac{{t}'-\mu }{\sigma } }[/math]

[math]\displaystyle{ {t}'=\ln (t) }[/math]

and:

[math]\displaystyle{ \begin{align} & \mu = & \text{scale parameter} \\ & \sigma = & \text{shape parameter} \end{align} }[/math]

where [math]\displaystyle{ 0\lt t\lt \infty }[/math] , [math]\displaystyle{ -\infty \lt \mu \lt \infty }[/math] and [math]\displaystyle{ 0\lt \sigma \lt \infty }[/math] .