[SOLVED] math graph statistic Assessment 1: Stochastic systems

Assessment 1: Stochastic systems
Implement a 33 discrete square grid. A turtle is placed in the bottom left corner (labelled 1) and moves in discrete time randomly across the square grid. The turtle is constrained to motion into east, west, north and south directions. Label the grid cells of the system like this:
789 456 123
Task 1: Find the transition probabilities between the cells such that the long-term (steady state) probability to be in any cell is the same. Represent the model as a Markov chain diagram (i.e. a directed graph) with the node labels corresponding to the cells numbers above. Label the graph arrows with the transition probabilities. Importantly, write the transition probabilities as fractions (i.e. 1/3) not as floats (i.e. 0.33333).
Deliverable: 1 graph representation. [3 Marks]
Task 2: Find transition probabilities between the cells such that the probability to be in the bottom row (cells 1,2,3) is 1/6. The probability to be in the middle row is 2/6. The probability of each cell in a row is the same. Represent the model as a Markov chain diagram (i.e. a directed graph) with the node labels corresponding to the cells numbers above. Label the graph arrows with the transition probabilities. Importantly, write the transition probabilities as fractions (i.e. 1/3) not as floats (i.e. 0.33333).
For task 1 and 2 you must use one of the methods discussed in the lectures.
Deliverable: 1 graph representation. [2 Marks] Task 3: Run the model from task 2 for 3 time steps. Based on 10000 repetitions report the probability and
standard deviation that at time step three (that is after three updates) the turtle is at cell 1,3,9. Deliverable: Report the three numbers in a Table of the following form:
Replace xxx, yyy, zzz by the relevant probabilities +/- a numerical estimate of your error. So, for example, zzz should be replaced by 0.1 0.004. An important part of the exercise is that you think about what an appropriate measure of error could be. In the row below, write down in 1 sentence, how you calculated the error. Ideally this should be a verbal explanation of what the number means. Stylistic examples are: Average over the last 15 lines. or Fano factor of the stochastic process calculated from the data. Code/pseudocode is not acceptable. If you put in a mathematical formula (I advise against), then make sure all symbols are explained. Well known statistical concepts such as variance, standard deviation, mean, or average do not need to be explained, but you need to say what the average/standard deviation/variance was taken over.
[3 Marks] Task 4: Run the model from task 2 once, but for a very large number of time-steps (say one million). Report
the probabilities and errors for the turtle to be in cell 1,3,9.
Deliverable: A table as in Task 3. [2 Marks]
Submit exactly 1 side A4 for all of the tasks 1-4. Point deduction if you fail to meet this specification. Submission is to the student administration office/reception. For deadline see the relevant SDS page.
Cell 1
Cell 3
Cell 9
xxx
yyy
zzz
Justification for your measure of variability.