# Generalized logistic distribution

I saw on wikipedia https://en.wikipedia.org/wiki/Generalized_logistic_distribution that when $$\alpha<\beta$$, generalized type IV logistic distribution can be written as:

$$\frac{\exp(-\alpha x)}{(\exp(-x)+1)^\beta}$$

But I'm wondering if the case $$\alpha\ge\beta$$ is studied?

• Could you point out where in that Wikipedia article it requires $\alpha\lt\beta$ for the Type IV distribution? I cannot see that anywhere. – whuber Apr 12 at 20:20
• If you reparametrize type IV into numerator power and denominator power you'll get this. @whuber – ZHU Apr 12 at 22:06
• Now I see the difference. But notice that this is supposed to be a probability density supported on $x\in\mathbb R:$ what is the integral of this pdf when $\beta \le \alpha$?? – whuber Apr 13 at 13:29