TEST BANK INSTRUCTOR’S SOLUTIONS MANUAL FOR INTRODUCTION TO STOCHASTIC PROCESSES WITH R BY ROBERT P. DOBROW

TEST BANK INSTRUCTOR’S SOLUTIONS MANUAL FOR INTRODUCTION TO STOCHASTIC PROCESSES WITH R BY ROBERT P. DOBROW Chapter 1 1.1 For the following scenarios identify a stochastic process Xt, t I , describing (i) Xt in context, (ii) state space, and (iii) index set. State whether the state space and index set are discrete or continuous. b) Xt is the student’s status at the end of year t. State space (discrete): S = {Drop Out, Frosh, Sophomore, Junior, Senior, Graduate}. Index set (discrete): I = {0, 1, 2, . . .}. c) Xt is the magnitude for an earthquake which occurs at time t. State space (contin- uous): (0, 10). Index set (continuous): [0, ∞). d) Xt is the circumference of the tree at location t. State space (continuous): (0, ). Index set (continuous): [0, 2] [0, 2]; or the x y coordinates of a location in the arboretum. e) Xt is the arrival time of student t. State space (continuous): [0, 60]. Index set (discrete): {1, 2, . . . , 30}. f) Xt is the order of the deck of cards after t shuffles. State space (discrete): Set of all orderings of the deck (52! elements). Index set (discrete): {0, 1, 2, . . .}. 1.2 A regional insurance company insures homeowners against flood damage. Half of their policyholders are in Florida, 30% in Louisiana, and 20% in Texas. a) Let A be the event that a claim is filed for flood damage. Then P (A) = P (A|F )P (F ) + P (A|L)P (L) + P (A|T )P (T ) = (0.03)(0.50) + (0.015)(0.30) + (0.02)(0.20) = 0.0235. b) P (T |A) = P (A|T )P (T )/P (A) = (0.02)(0.20)/0.0235 = 0.17. 1.3 Let B1, . . . , Bk be a partition of the sample space. For events A and C, prove the law of total probability for conditional probability. Σ Σ P (A ∩ B ∩ C) P (B ∩ C) i=1 P (A|Bi ∩ C)P (Bi|C) = i=1 k i P (Bi ∩ C) i P (C) k = Σ P (A ∩ Bi ∩ C) = 1 Σ P (A ∩ B ∩ C) = P (A ∩ C) = P (A C). P (C) 1.4 See Exercise 1.2. Among policyholders who live within 5 miles of the coast, 75% live in Florida, 20% live in Louisiana, and 5% live in Texas. Suppose a policyholder lives within 5 miles of the coast. Use the law of total probability for conditional probability to find the chance they will file a claim for flood damage next year.