\textbf{Problem 6}

Company A owns multiple factories $i ( i = 1,2 , \cdots )$. Suppose that the probability of producing defective goods in a factory $i$ is $P _ { i }$, and that $N _ { i }$ goods are randomly sampled and shipped from the factory. Here, $P _ { i }$ is sufficiently small, and each factory does not affect any other.

I. Show the probability $f ( i , k )$, which is the probability of $k$ defective goods existing within $N _ { i }$ goods shipped from a factory $i$. Here, $k$ is a non-negative integer.

II. Show that $f ( i , k ) \rightarrow \frac { e ^ { - \lambda _ { i } } \lambda _ { i } ^ { k } } { k ! }$ when $N _ { i } \rightarrow \infty$. Here, when calculating the limit of $f ( i , k ) , \lambda _ { i }$ is a constant, where $\lambda _ { i } = N _ { i } P _ { i }$.

In the following questions, assume that $f ( i , k ) = \frac { e ^ { - \lambda _ { i } } \lambda _ { i } ^ { k } } { k ! }$.

III. Suppose that goods are shipped from two factories as shown in Table 1. Find the probability of two defective goods being contained within all shipped goods.

\begin{center}
\begin{tabular}{ c c c }
\hline
\begin{tabular}{ c }
Factory number \\
$( i )$ \\
\end{tabular} & \begin{tabular}{ c }
Probability of defectiveness \\
$\left( P _ { i } \right)$ \\
\end{tabular} & \begin{tabular}{ c }
Number of shipped goods \\
$\left( N _ { i } \right)$ \\
\end{tabular} \\
\hline
1 & 0.01 & 500 \\
\hline
2 & 0.02 & 300 \\
\hline
\end{tabular}
\end{center}

IV. Find the probability of $k$ defective goods being contained within all shipped goods under the same conditions as in Question III.

V. Suppose that $P _ { i } = 0.001 i$ in five factories $i ( i = 1,2,3,4,5 )$ and the same number ($N _ { c}$) of goods are shipped from all these factories.

Find the maximum value of $N _ { c }$ for which the expected number of defective goods out of all shipped goods is equal to or less than 3.