Problem 6Company 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{tabular}{ c } Factory number |
| $( i )$ |
&
| Probability of defectiveness |
| $\left( P _ { i } \right)$ |
&
| Number of shipped goods |
| $\left( N _ { i } \right)$ |
\hline 1 & 0.01 & 500 \hline 2 & 0.02 & 300 \hline \end{tabular}
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.