Template:Loglogistic distribution characteristics

Distribution Characteristics
For $$\sigma >1$$ :


 * •	 $$f(T)$$ decreases monotonically and is convex. Mode and mean do not exist.

For $$\sigma =1$$ :


 * •	 $$f(T)$$ decreases monotonically and is convex. Mode and mean do not exist. As  $$T\to 0$$,  $$f(T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.$$
 * •	As $$T\to 0$$ ,  $$\lambda (T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.$$

For $$0<\sigma <1$$ :


 * •	The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
 * •	The $$pdf$$  starts at zero, increases to its mode, and decreases thereafter.
 * •	As $$\mu $$  increases, while  $$\sigma $$  is kept the same, the  $$pdf$$  gets stretched out to the right and its height decreases, while maintaining its shape.
 * •	As $$\mu $$  decreases,while  $$\sigma $$  is kept the same, the  ..  gets pushed in towards the left and its height increases.
 * •	 $$\lambda (T)$$ increases till  $$T={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}$$   and decreases thereafter.  $$\lambda (T)$$  is concave at first, then becomes convex.