Question
Ladislaus Bortkiewicz was a Russian economist and statistician who published a book entitled "The Law of Small Numbers." In his book he showed that the number of soldiers in the Prussian cavalry killed by being kicked by a horse each year in each of 14 cavalry corps over a 20 -year period (1875-1894) followed a Poisson distribution. ${ }^{10}$ The data summary follows.$$\begin{array}{c|r}\text { Number of deaths } & \text { Frequency } \\\hline 0 & 144 \\1 & 91 \\2 & 32 \\3 & 11 \\4 & 2\end{array}$$a. Find the mean number of deaths per year per cavalry unit. [HINT: Use the grouped formula given in Exercise 21 of the "On Your Own" Exercises in Chapter $2 .$b. Use the result of part a and the Poisson distribution to find the probability of exactly one death per unit per year.c. Find the probability of at most two deaths per year.d. How do the probabilities in parts $\mathrm{b}$ and $\mathrm{c}$ compare to the observed relative frequencies in the table?
Step 1
We can use the grouped formula for this. The formula is given by $\mu = \frac{\Sigma x_i f_i}{n}$, where $x_i$ is the number of deaths, $f_i$ is the frequency of each number of deaths, and $n$ is the total number of observations. Show more…
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In 1898 , Ladislaus von Bortkiewicz published The Law of Small Numbers, in which he demonstrated the power of the Poisson probability law. Before his publication, the law was used exclusively to approximate binomial probabilities. He demonstrated the law's power, using the number of Prussian cavalry soldiers who were kicked to death by their horses. The Prussian army monitored 10 cavalry corps for 20 years and recorded the number $X$ of annual fatalities because of horse kicks for the 200 observations. The following table shows the data: $$\begin{array}{ll}\hline \text { Number of } & \text { Number of Times } \boldsymbol{x} \\\text { Deaths, } \boldsymbol{x} & \text { Deaths Were Observed } \\\hline 0 & 109 \\\hline 1 & 65 \\\hline 2 & 22 \\\hline 3 & 3 \\\hline 4 & 1 \\\hline\end{array}$$ (a) Compute the proportion of years in which there were 0 deaths, 1 death, 2 deaths, 3 deaths, and 4 deaths. (b) From the data in the table, what was the mean number of deaths per year? (c) Use the mean number of deaths per year found in part (b) and the Poisson probability law to determine the theoretical proportion of years that 0 deaths should occur. Repeat this for $1,2,3,$ and 4 deaths. (d) Compare the observed proportions to the theoretical proportions. Do you think the data can be modeled by the Poisson probability law?
Discrete Probability Distributions
The Poisson Probability Distribution
Prussian Army In $1898,$ Ladislaus von Bortkicwicz published The Law of Small Numbers, in which he demonstrated the power of the Poisson probability law. Before his publication, the law was used exclusively to approximate binomial probabilities. He demonstrated the law's power, using the number of Prussian cavalry soldiers who were kicked to death by their horses. The Prussian army monitored 10 cavalry corps for 20 years and recorded the number $X$ of annual fatalities due to horse kicks for the 200 observations. The following table shows the data: (a) Compute the proportion of years in which there were 0 deaths, 1 death, 2 deaths, 3 deaths, and 4 deaths. (b) From the data in the table, what was the mean number of deaths per year? (c) Use the mean number of deaths per year found in part (b) and the Poisson probability law to determine the theoretical proportion of years that 0 deaths should occur. Repeat this for $1,2,3,$ and 4 deaths. (d) Compare the observed proportions to the theoretical proportions. Do you think the data can be modeled by the Poisson probability law?
12. Deaths from Horse Kicks A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army berween 1875 and 1894. Data for 14 corps were combined for the 20 -year period, and the 280 corps-years induded a total of 196 deachs. After finding the mean number of deachs per corps-ycar, find the probability that a randomly sclected corps-ycar has the following numbers of deaths. a.o b. 1 c. 2 d. 3 e. 4 The actual results consisted of these frequencics: 0 deaths (in 144 corps-ycars) il death (in 91 corps-years) years). Compare the actual results to those expected by using the Poisson probabilitics. Docs the Poisson distribution serve as a good device for predicting the actual results?
Poisson Probability Distributions
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