It is known that a certain disease has affected 10% of a population. In a random sample of 50 patients typical of the disease group who are exposed to a new treatment, we observe that 12 patients were hospitalized in a year. Let μ be the rate of population that need hospitalization. Assume that
μ ~ Gamma (0.1, 2) and f (x | μ) ~ Poi (50m):
Given that 0.24 is an observation from f(x | μ), find the Bayesian estimator of μ (that is, obtain E(μ | x)).