SEEM 4480 Decision Methodology and Applicatons Xuedong He, Fall 2017 Homework Set 5 (Due Apr. 13) Assignment 1. (a) Consider a two-act problem with linear value where the corresponding utility functions are given by u(a1 , s) = 840s 10, u(a2 , s) = 360s + 200, where s follows beta distribution with parameter ( = 3; v = 25). What is the optimal action? What is the EVPI? (b) Suppose a loss function is given in the following form l(a, s) = 1200(a s)2 , where s is the random state which follows beta distribution with parameter ( = 4; v = 21). What is the best point estimate for s? What is the EVPI? 2. Weekday lunch demand for spicy black bean burritos at the Kiosk, a local snack bar, is approximately Poisson distributed with mean of 22. The Kiosk charges $4.60 for each burrito, which are made before the lunch crowd arrives. Each burrito costs the Kiosk $2.00. Kiosk management is very sensitive about the quality of food they serve; thus they maintain a strict \"No Old Burrito\" policy, so any burrito left at the end of the day is disposed of. (a) Suppose burrito customers buy their snack elsewhere if the Kiosk is out of stock. How many burritos should the Kiosk make to maximize its expected profit? (b) Suppose any customer unable to purchase a burrito will buy Pop-Tarts instead, which never run out of stock. Each Pop-Tart sells for $0.75 and costs the Kiosk $0.25. How many burritos should the Kiosk make in this case to maximize its expected profit? 3. Let the parameter p of a Bernoulli process be given a beta distribution with parameters = 7 and v = 18. Give the probabilities of the following events: (a) S1 . (b) S2 given F1 . (c) S3 given F1 F2 . (d) S4 given F1 F2 F3 . (e) S5 given F1 F2 S3 F4 . 1 SEEM 4480 Decision Methodology and Applicatons Xuedong He, Fall 2017 4. Consider a sequence of samples zi , i 1 that are i.i.d. and follow normal distribution with mean 0 and variance 1/p given parameter p. Here, p is referred to the precision of the normal random variable. Suppose that the precision is unknown and modeled by a random variable p. The distribution of p is Gamma1 with parameters (, ). Given the observations zi = zi , i = 1, . . . , n, find the posterior distribution of p. 2