\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

In the branching chain obtained when considering the disappearance
of family lines suppose that the number of male offspring of any
male individual has a binomial distribution with $n = 10$ and $p =
\frac{1}{4}$, independently of the number of offspring of any
other male. Find the 1-step transition probability $p_{jk}$ for
the total number of males in a generation.



\vspace{.25in}

{\bf Answer}

Let $Z_i$ be the number of male children produced by the $i$-th
male in a generation.

Then $Z_i \sim B\left(10, \frac{1}{4}\right)$

If in generation $m$ there are $j$ males then the number of males
in generation $m+1$ is $X_{m+1} = Z_1 + ... + Z_j$

So $X_{m+1}$ is a sum of $j$ i.i.d. binomial r.v.'s and so $
X_{m+1} \sim B\left(10j, \frac{1}{4}\right)$ so

$p_{jk} = \left\{ \begin{array}{cc} \left(\begin{array}{c} 10j \\
k \end{array} \right) \ds\left( \frac{1}{4} \right)^k \left(
\frac{3}{4} \right)^{10j - k} & {\rm if\ \ } k \leq 10j \\ 0 &
{\rm otherwise} \end{array} \right.$






\end{document}