Bayesian Analysis of a Mixed Poisson Process
Consider a scenario where the number of events, denoted by \(x\), follows a Poisson distribution with a rate parameter \(\lambda\). However, instead of \(\lambda\) being a fixed value, it is itself a random variable following a Gamma distribution with parameters \(\alpha\) and \(\beta\). This hierarchical model is defined as follows: – Level 1: Conditional distribution […]
Part b
Assume that initial trials suggest a prior defect probability \(p_0\). Using Bayes’ theorem, explain how this prior probability \(p_0\) is updated with new trial data to produce a posterior probability of defect. Suppose in a new sample of \(m\) items, \(k\) defective items are found. Detail the application of Bayes’ theorem in this scenario.