11.Compute the 4-step transition probability Matrix (P) 12.Compute the 8-step transition probability Matrix (P) 13. Compute the 16-step transition probability Matrix (Pl) 14. Write down the initial transition probability vector from the fair state to all other possible states (a). 15. Write down the transition probability vector from the fair state to all other possible states after 1 season (a'). 16.Compute the transition probability vector from the fair state to all other possible states after 2 season (a). 17. Compute the transition probability vector from the fair state to all other possible states after 4 season (af). 18. Compute the transition probability vector from the fair state to all other possible states after 8 season (a). 19.Compute the transition probability vector from the fair state to all other possible states after 16 season (a'). Page 1 of 2 20.Use excel Spreadsheet to compute the transition vector from the fair state to all other possible states over infinite number of seasons (select a number of season geater than 15, constuct a table as the one presented in the slide shown below). 21.Compute the steady state transition probability vector from the fair (or poor) state to all other possible states. 22.Compute the mean first return time to every state { X=j= good, fair, poor ). 23.The garden needs 2 bags of fertilizer if the soil is good. The amount is increased by 25% if the soil is fair and 60% if the soil is poor. The cost of the fertilizer is $50 per bag, Estimate the seasonal expected cost of fertilizer. 24.Compute how many seasons it takes on average to pass from good to fair soil, and from poor to fair soil (the passage from states good and poor to state fair). 25. Compute how many seasons it takes on average to pass from good to poor soil, and from fair to poor soil (the passage from states good and fair to state poor). Due 5/1/2022 @11:59 pm Every year, during the March-through-September growing season, a gardener uses a chemical test to check soil condition. Depending on the outcome of the soil test, productivity for the new season can be one of three states: (1) good, (2) fair, and (3) poor. Over the year, the gardener has observed that last year's soil condition impacts current year's productivity. If this year's soil condition is good, there is a 0.5 chance it will not change next year, and a 0.3 chance it will be fair. If this year's soil condition is fair, there is a 0.5 chance it will not change next year, and a 0.4 chance it will be poor. If this year's soil condition is poor, there is a 0.6 chance it will not change next year, and a 0.3 chance it will fair. 1. Define the proper states of this system. 2.Construct the set of state space (S = {......... 3. Is this a stochastic system process, and why?. 4. Construct the state transition Diagram for the system. 5.Check (with explaination) for Markovian property (Markov Process) 6.Construct the one step transition matrix (P) 7. Comment on the one step transition matrix (Pl) ( hint: Row's sum, Matrix Dimension, values, etc.) 8.Can you describe the system as a discrete-time Markov chain (Markov chain), why?. 9. Write the Initial probability matrix (hint: P) 10.Compute the 2-step transition probability Matrix (P) 11. Compute the 4-step transition probability Matrix (P) 12.Compute the 8-step transition probability Matrix (P) 13.Compute the 16-step transition probability Matrix (Pl) 14. Write down the initial transition probability vector from the fair state to all other possible states (a'). 15. Write down the transition probability vector from the fair state to all other possible states after 1 season (a'). 16.Compute the transition probability vector from the fair state to all other possible states after 2 season (a), 17. Compute the transition probability vector from the fair state to all other possible states after 4 season (a). 18. Compute the transition probability vector from the fair state to all other possible states after 8 season (a). 19.Compute the transition probability vector from the fair state to all other possible states after 16 season (a). Page 1 of 2 20. Use excel Spreadsheet to compute the transition vector from the fair state to all other possible states over infinite number of seasons (select a number of season geater than 15, constuct a table as the one presented in the slide shown below). 21.Compute the steady state transition probability vector from the fair (or poor) state to all other possible states. 22.Compute the mean first return time to every state Hi j= good, fair, poor