Question 1 ( 50 points) A physical-fitness center has found that, of those members who perform highimpact exercising (1), will either repeat high-impact exercising, or do medium-impact exercising (2), lowimpact exercising (3) cardio exercise (4) in their next visit to the fitness center. To determine the exercise habits of its member, the fitness center collected some data from its members through a survey and prepared the following table: a) (10 points) How many members answered to survey? How many how those answered the survey did Cardio in their current exercise? b) (10 points) Identify the epochs of the Discrete Time Stochastic Process. c) (10 points) Explain why this example can be modeled as a Discrete Time Markov Chain? d) (20 points) Explain the Stationarity assumption in this context of the above situation and provid a situation where the stationarity assumption is not reasonable. Question 2 ( 50 points) Consider the situation given in question 1. a) (10 points) Define Xn and write down the state space S. b) (10 points) Write down one-step transition matrix P. c) (10 points) Write down the mathematical expression for the following probability: The probability that the athlete will do a cardio exercise after making low impact exercise twice. Find the probability of it. d) (10 points) What is the probability that after a medium impact exercise the athlete will train twice on cardio training. Represent this probability as a mathematical expression. Discuss which assumptions are needed to calculate this probability. e) (10 points) Provide a walk of three transitions that includes 4 different exercises and calculate the probability of this walk. Question 1 ( 50 points) A physical-fitness center has found that, of those members who perform highimpact exercising (1), will either repeat high-impact exercising, or do medium-impact exercising (2), lowimpact exercising (3) cardio exercise (4) in their next visit to the fitness center. To determine the exercise habits of its member, the fitness center collected some data from its members through a survey and prepared the following table: a) (10 points) How many members answered to survey? How many how those answered the survey did Cardio in their current exercise? b) (10 points) Identify the epochs of the Discrete Time Stochastic Process. c) (10 points) Explain why this example can be modeled as a Discrete Time Markov Chain? d) (20 points) Explain the Stationarity assumption in this context of the above situation and provid a situation where the stationarity assumption is not reasonable. Question 2 ( 50 points) Consider the situation given in question 1. a) (10 points) Define Xn and write down the state space S. b) (10 points) Write down one-step transition matrix P. c) (10 points) Write down the mathematical expression for the following probability: The probability that the athlete will do a cardio exercise after making low impact exercise twice. Find the probability of it. d) (10 points) What is the probability that after a medium impact exercise the athlete will train twice on cardio training. Represent this probability as a mathematical expression. Discuss which assumptions are needed to calculate this probability. e) (10 points) Provide a walk of three transitions that includes 4 different exercises and calculate the probability of this walk