A machine is built to make mass-produced items. Each item made by the machine has a probability θ of being defective. Given the value of θ, the items are independent of each other, where θ is unknown and would like to estimate. Suppose θ has for prior distribution a Beta(α,β) distribution, where α > 0 and β > 0. The machine is tested by producing items until the first defective occurs. Suppose that the first 12 items are not defective but the y= 13th item is defective. (a) Write down the likelihood function for θ and find the MLE of θ. (b) Given the observed data y = 13, what is the posterior distribution of θ, p(θ |y= 13)? Take α= 1 and β = 19. (c) What are the parameters of the posterior distribution? (d) What is the posterior mean for θ? (e) What is the posterior standard deviation?