4. Let Y1,Y2,...,Yn have a Poisson distribution with mean . Let the prior distribution of be a Gamma distribution with shape parameter and scale parameter .

(a) Show the posterior distribution of belongs to the Gamma family.

(b) Determine the parameters ( and ) of the posterior distribution.

(c) Download the data set poissondata.csv from Canvas and read the data set into R. Use the sample data and your answers above to find a 95% credible interval for the parameter if = 5 and = 2.

Poissondata:

Values 11 14 14 10 17 12 8 15 10 11 13 16 13 14 16 12 10 12 19 10 11 4 9 10 12