I am interested in exploring the probability distribution given by:- $$\mathbb{P}(X\ge x)=F(x)=\alpha^{\beta^x}$$ with probability density function, $$f(x)=F'(x)=\log(\alpha)\log(\beta)\beta^x\alpha^{\beta^x}=\log(\alpha)\log(\beta)\beta^xF(x)$$ The parameter $\alpha$ slides the curve along the axis, with $\alpha=\frac12$ giving the distribution a median of zero. The parameter $\beta$ controls how spread out the distribution is, with $\beta=\frac13$ it looks a bit like a slanted normal distribution. The function $F(x)$ has the interesting property that raising it to a power simply translates it along the x-axis:- $$F^n(x)=(\alpha^{\beta^x})^n=\alpha^{n\beta^x}=\alpha^{\beta^{x+\gamma}}=F(x+\gamma)$$ where $\gamma=\frac{\log n}{\log\beta}$
This is very helpful as I have been trying to perform calculations involving maximum order statistics, $X_{Max}=\max\limits_{i=1}^n(X_i)$:- $$\mathbb{P}(X_{Max}\ge x)=(\mathbb{P}(X\ge x))^n$$ I am however having difficulty in solving the integrals to work out the mean and standard deviation of this distribution.
Does anyone know if this distribution has a name and where I can find further information about it?
