A Tetration Distribution

Abstract

We introduce a one-parameter family of continuous distributions on the unit interval whose cumulative distribution functions are given by the odd-height power towers \(x\uparrow\uparrow n\) (\(n=1,3,5,\dots\)). The associated densities admit a simple recursive formula obtained by logarithmic differentiation. We prove that every such density is strictly positive on \((0,1)\) and integrates to one, thereby establishing that the construction yields a genuine probability distribution. Numerical investigation reveals a characteristic peak--dip--rise shape: a sharp mode lying just below \(e^{-e}\) for all examined heights, followed by a trough and a gradual recovery toward density value~1 at the right endpoint. We then identify the limiting distribution as the height tends to infinity through odd integers.