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Question 2 (60 points) German automobile manufacturers are concerned about the competition of foreign cars in the domestic market. Volkswagen, the producer of famous BMW and Porsche brands, views its survival upon recapturing market share it lost to Japanese cars over last 10 years. To determine its present market share a survey of 1,000 car owners were conducted: 500 respondents owned a Mercedes car, 200 respondents owned a Volkswagen car, and the remaining 300 respondents owned Japanese brands such as Toyota, Honda, Nissan, etc. The results of the survey is summarized below: 302 of present Mercedes car owners said they would probably buy another Mercedes car for their next purchase, 80 of them said that they would buy a Volkswagen brand, and the remaining said they would buy a Japanese car for their next purchase. 97 of present Volkswagen car owners said they would probably buy another Volkswagen car for their next purchase, 90 of them said that they would buy a Mercedes brand, and the remaining said they would buy a Japanese car for their next purchase. 250 of present Japanese car owners said that they were very happy with Japanese brands, and 184 of them said they would probably buy a Japanese brand for their next purchase. The remaining respondents were equally likely to buy a Mercedes or a Volkswagen brand. a) (10 points) Explain if we can model the above brand switching problem as a discrete-time stochastic process. b) (10 points) Define Xn and write down the state space, S. c) (10 points) Is {Xn, n=0,1,2,...} a Markov chain? Why or why not? Explain. d) (10 points) What would be the appropriate observation points (epochs) of the owners purchase behavior? Explain. e) (10 points) Prepare a table to summarize a the results of the survey. f) (10 points) Write down the one-step transition probability P matrix. Question 2 (60 points) German automobile manufacturers are concerned about the competition of foreign cars in the domestic market. Volkswagen, the producer of famous BMW and Porsche brands, views its survival upon recapturing market share it lost to Japanese cars over last 10 years. To determine its present market share a survey of 1,000 car owners were conducted: 500 respondents owned a Mercedes car, 200 respondents owned a Volkswagen car, and the remaining 300 respondents owned Japanese brands such as Toyota, Honda, Nissan, etc. The results of the survey is summarized below: 302 of present Mercedes car owners said they would probably buy another Mercedes car for their next purchase, 80 of them said that they would buy a Volkswagen brand, and the remaining said they would buy a Japanese car for their next purchase. 97 of present Volkswagen car owners said they would probably buy another Volkswagen car for their next purchase, 90 of them said that they would buy a Mercedes brand, and the remaining said they would buy a Japanese car for their next purchase. 250 of present Japanese car owners said that they were very happy with Japanese brands, and 184 of them said they would probably buy a Japanese brand for their next purchase. The remaining respondents were equally likely to buy a Mercedes or a Volkswagen brand. a) (10 points) Explain if we can model the above brand switching problem as a discrete-time stochastic process. b) (10 points) Define Xn and write down the state space, S. c) (10 points) Is {Xn, n=0,1,2,...} a Markov chain? Why or why not? Explain. d) (10 points) What would be the appropriate observation points (epochs) of the owners purchase behavior? Explain. e) (10 points) Prepare a table to summarize a the results of the survey. f) (10 points) Write down the one-step transition probability P matrix