[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Particles are subject to collisions that cause them to split into

two parts with each part a fraction of the parent. Suppose that this fraction is uniformly distributed between 0 and 1. Following a single particle through several splittings we obtain a fraction of the original particle [math]Z_n = X_1 \cdot X_2 \cdot\dots\cdot X_n[/math] where each [math]X_j[/math] is uniformly distributed between 0 and 1. Show that the density for the random variable [math]Z_n[/math] is

[[math]] f_n(z) = \frac 1{(n - 1)!}( -\log z)^{n - 1}. [[/math]]

Hint: Show that [math]Y_k = -\log X_k[/math] is exponentially distributed. Use this to find the density function for [math]S_n = Y_1 + Y_2 +\cdots+ Y_n[/math], and from this the cumulative distribution and density of [math]Z_n = e^{-S_n}[/math].