CDF for Loglogistic distribution

What is the Cumulative Distribution Function for Loglogistic distribution? I only found the Probability Density Function in the following link: https://www.mathworks.com/help/stats/loglogistic-distribution.html.

[Walter Roberson]

https://en.m.wikipedia.org/wiki/Log-logistic_distribution

[Zhengpu Chen]

Thank you for the reply. The expression of the function in Wiki and the one in Matlab documentation https://www.mathworks.com/help/stats/loglogistic-distribution.html seem to be not the same. I don't know if they are equivalent.

Answer 1

[David Goodmanson]

Hello ZC,

They are equivalent. In Matlab

cdf =               e^z / (1+e^z)            % the missing one
pdf = (1/(sigma*x))*e^z / (1+e^z)^2
where
z = (log(x)-mu)/sigma

When you make the substitution (as mentioned in wikipedia, where they use s in place of sigma)

beta = 1/sigma        e^mu = alpha
then
e^z = (x/alpha)^beta

and for wiki,

cdf =              (x/alpha)^beta)   /(1+(x/alpha)^beta) 
pdf = (beta/alpha)*(x/alpha)^(beta-1)/(1+(x/alpha)^beta)^2