- 1 The Proportion Of Defective Items In A Large Shipment Is Unknown But A Beta Prior Probability Density Function Of Th 1 (91.5 KiB) Viewed 72 times
1. The proportion of defective items in a large shipment is unknown, but a beta prior probability density function of the form 1 π(θ) : B(α, ß) With corresponding mean and variance, α 0 <0 <1 and B(a, B) j0“1(1–0)ß-1, E(0) = a + ß and var (0) = Γ(α)Γ(β) [(a + B) αβ (a + b)²(a + B + 1) a) Given that the prior mean E (0) and standard deviation (0) are both 10-², (i) use the formula of E(0) to find an expression for ẞ in terms of a. (3 marks) (ii) Substitute this into the formula of var (0) to evaluate the hyperparameters a and B. (5 marks) b) Given that 100 items are selected at random from the shipment and 3 of these are found to be defective. Determine the posterior probability density function of 0 [Hints: observations have the Binomial distribution] c) Find the Bayesian estimate of 0 under the quadratic loss function. (9 marks) (4 marks)