\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}
{\bf Question}
A transistor in a machine fails has to be replaced on average
twice a week according to a Poisson process. Replacement
transistors are bought annually and kept in store for use during
the year. How many transistors should be bought to ensure only a
5\% chance of running out of replacement during the year?
\vspace{.25in}
{\bf Answer}
Let $N(t)$ be the number of breakdowns in $t$ weeks. This has a
Poisson distribution with parameter $2t$.
We want to buy $n$ components, where $$P(N(52)>n) \leq 0.05$$
$\ds N(52)\sim P(104) \approx N(104,104) \, (= N_p)$
$\ds P(N_p > n) = P \left( Z > \frac{n-104}{\sqrt{104}} \right)
\leq 0.05$
so $\ds \frac{n-104}{\sqrt{104}} \geq 1.645 \hspace{.2in} n \geq
121$
\end{document}