PROBLEM #3: Discrete data. Table below gives the number of fatal accidents and d

Question

PROBLEM #3: Discrete data. Table below gives the number of fatal accidents and deaths on scheduled airline flights per year over a ten-year period. We use these data as a numerical example for fitting discrete data models Year Fatal Passenger deaths Death accidents rate 1976 24 1977 25 1978 31 197931 1980 22 1981 21 1982 26 1983 20 1984 16 1985 22 734 516 754 877 814 362 764 809 223 1066 0.19 0.12 0.15 0.16 0.14 0.06 0.13 0.13 0.03 0.15 Worldwide airline fatalities, 1976-1985 Death rate is passenger deaths per 100 million passenger miles 5pt (a) Assume that the numbers of fatal accidents in each year are inde- pendent with a Poisson() distribution. Set a prior distribution for and determine the posterior distribution based on the data from 1976 through 1985, Under this model, give a 95% predictive interval for the number of fatal accidents in 1986 5pt (b) Assume that the numbers of fatal accidents in each year follow inde- endent Poisson distributions with a constant rate and an exposure in each year proportional to the number of passenger miles flown. Set a prior distribution for and determine the posterior distribution based on the data for 1976-1985. (Estimate the number of passenger miles flown in each year by dividing the appropriate columns of the Table and ignoring round-off errors.) Give a 95% predictive interval for the number of fatal accidents in 1986 under the assumption that 8 ×10 passenger miles are flown that year.

Explanation / Answer

a) Prior Distribution:- Let X=Number of passenger, Y=Expected accident rate per passenger

The modal for the data is y/x, Y~Poisson(xY) By using Gamma(0,0) prior distribution for Y.

Then the posterior distribution for Y= y/Y~Gamma(10y,10x)= Gamma(734,0.19).

Y, the predictive distribution for y is Poisson (xY)=Poisson(1066,0.15).

So, Here are two methods of obtaining a 95% posterior interval for y. For the number of fatal accidents in 1986.

b) With the help of prior distribution of x we show the posterior distribution for y, But in this case the predictive distribution for y is Poisson (xY)=(8X1011y)

c) In the above two cases the poisson modal seen more reasonable in 1st case because in this case value is used from the table which is more reliable.

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