During the latter days of World War II, the Germans developed flying rocket bombs. These bombs were used to attack London. Allied military intelligence didn't know whether these bombs were fired at random or had a sophisticated aiming device. To determine the answer, they used the Poisson distribution.
To assess the accuracy of these bombs, London was divided into 576 square regions. Each region was \(\frac{1}{4}\) square kilometer in area. They then compared the number of actual hits with the theoretical number of hits by using the Poisson distribution. If the values in both distributions were close, then they would conclude that the rockets were fired at random. The actual distribution is as follows:

1. Using the Poisson distribution, find the theoretical values for each number of hits. In this case, the number of bombs was 535, and the number of regions was 576 . So
\[\lambda=\frac{535}{576} \approx 0.929\]
For 3 hits,
\[\begin{aligned}
P(X) & =\frac{e^{-\lambda} \cdot \lambda^{X}}{X!} \\
& =\frac{(2.7183)^{-0.929}(0.929)^{3}}{3!} \approx 0.0528
\end{aligned}\]
Hence, the number of hits is \((0.0528)(576)=30.4128\).
Complete the table for the other number of hits.

2. Write a brief statement comparing the two distributions.
3. can you conclude that the rockets were fired at random?

Transcribed Image Text:

Hits Regions 0 93 229 213 41 35 7