The number of defects per 10 meters of cloth produced by a weaving machine has the Poisson distribution with mean . You examine 100 meters of cloth produced by the machine and observe 71 defects.

(a) Your prior belief about is that it has mean 6 and standard deviation 2. Find a \(\operatorname{gamma}(r v)\) prior that matches your prior belief.

(b) Find the posterior distribution of given that you observed 71 defects in 100 meters of cloth.

(c) Calculate a \(95 \%\) Bayesian credible interval for .