[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]

Consider a branching process such that the number of offspring of a

parent is [math]j[/math] with probability [math]1/2^{j + 1}[/math] for [math]j = 0[/math], 1, 2, \ldots.

• Using the results of Example show that the probability that there are [math]j[/math] offspring in the [math]n[/math]th generation is
  [[math]] p_j^{(n)} = \left \{ \begin{array}{ll} \frac{1}{n(n + 1)} (\frac {n}{n + 1})^j, & \mbox{if $ j \geq 1$}, \\ \frac {n}{n + 1}, & \mbox{if \ltmath\gt j = 0[[/math]]
  }.\end{array}\right. </math>
• Show that the probability that the process dies out exactly at the [math]n[/math]th generation is [math]1/n(n + 1)[/math].
• Show that the expected lifetime is infinite even though [math]d = 1[/math].