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\newcommand{\ds}{\displaystyle}
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{\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 $$P(N(52)&gt;n) \leq 0.05$$

$\ds N(52)\sim P(104) \approx N(104,104) \, (= N_p)$

$\ds P(N_p &gt; n) = P \left( Z &gt; \frac{n-104}{\sqrt{104}} \right)
\leq 0.05$

so $\ds \frac{n-104}{\sqrt{104}} \geq 1.645 \hspace{.2in} n \geq
121$

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