# The probability distribution $\alpha^{\beta^x}$

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?

-
The Gumbel distribution is of a similar form and widely used in extreme value analysis. – Eckhard Jan 13 '13 at 15:49
Many thanks, searching for "Gumbel" has found me lots of useful material. – Shard Jan 13 '13 at 16:02
If you're happy with my comment do you want me to post it as an answer so that the question can be marked answered? Or are you still looking for more detailed explanations of the evaluation of your integrals? – Eckhard Jan 13 '13 at 17:38
I think the wikipedia page on Gumbel has enough to answer any further questions I have on the distribution, so posting your comment as an answer would be fine. Thanks :) – Shard Jan 13 '13 at 18:10

## 1 Answer

In order to allow the question to be marked as answered I'm re-posting my comment as an answer.

The Gumbel distribution has a form similar to your distribution and is widely used in extreme value analysis.

-