1. The Russian mathematician Ladislaus Bortkiewicz noted that the Poisson distribution would apply to low frequency events in a large population, even when the probabilities for individuals in the population varied. In a famous example he showed that the number of deaths by horse kick per year in the cavalry corps of the Russian army follows the Poisson distribution. The following data is reproduced from Hoel (1984). y( deaths) 0 1 2 3 4 n(y) (frequency) 109 65 22 3 1 a)Suppose a prior uniform distribution is used for ?. Find the corresponding posterior mean, variance, median. b) Suppose that the prior g(?) = ?^-½ is used for ?. Find the corresponding posterior mean, variance, median.
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### Part a) Uniform Prior Distribution for \(\lambda\) #### Show more…
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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?
Discrete Probability Distributions
The Poisson Probability Distribution
In Exercise 16.11, we found the posterior density for $\lambda$ the mean of a Poisson-distributed population. Assuming a sample of size $n$ and a conjugate gamma $(\alpha, \beta)$ prior for $\lambda$, we showed that the posterior density of $\lambda | \sum y_{i}$ is gamma with parameters $\alpha^{*}=\Sigma y_{i}+\alpha$ and $\beta^{*}=\beta /(n \beta+1)$. If a sample of size $n=25$ is such that $\sum y_{i}=174$ and the prior parameters were $(\alpha=2, \beta=3),$ use the applet Gamma Probabilities and Quantiles to find a $95 \%$ credible interval for $\lambda$.
Introduction to Bayesian Methods for Inference
Bayesian Credible Intervals
7. A large tank of fish from a hatchery is being delivered to a lake. The hatchery claims that the mean length of fish in the tank is 15 inches, and the standard deviation is 2 inches. A random sample of 36 fish is taken from the tank. a. 5 pts. What is the mean and the standard deviation of the sample mean length of these fish? b. 3 pts. What is the distribution of the sample mean length of the randomly selected 36 fish? Write down the distribution by correct notations and related parameter values. Suppose we don't know the mean and standard deviation of the fish in the tank. A sample of 36 fish from the tank has a mean of 13 inches and a sample standard deviation of 1.8 inches. c. 5 pts. Make a 99% confidence interval for the population mean length of these fish in the tank. d. 4 pts. Use one sentence to interpret the meaning of the 95% confidence interval in c).
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