I need initial post and 2 responses to classmates.
Modeling Real-World Probability Over Time
Suppose you and a team of coworkers are planning for the likelihood of two outcomes that may affect your company’s business in the future. You are tasked with modeling the interrelationship between two states, or outcomes, with a Markov system (i.e., state-transition diagram). Some possibilities would be the proportion of users who are paying for your service versus those who are not, devices that are produced to specification codes versus those that turn out defective, or kiosks that are working properly versus those that have started to malfunction.
Post 1: Initial Response
Develop a hypothetical scenario in which you and your team members are going to examine the likelihood that someone or something will be observed in one of two states at any point in time. As you develop your scenario, carefully address all of the following:
- In a brief narrative introducing your scenario, clearly identify two states for your scenario. Be sure to describe what the states are and define them as State A versus State B.
- Using the illustration provided as a guide, clearly present the probabilities associated with each of the four possible transitions, by labeling each of the transitions (indicated by the arrows) with an appropriate value for the probability they will occur (i.e., these are your own hypothetical estimates). Be sure these values adhere to probability norms regarding what range they may fall within and how they combine or relate with their paired probability. Share your complete state-transition diagram with a visual.
- Propose an initial distribution vector, v, which gives a hypothetical estimate of what proportion is initially observed in State A and State B. This should be reported as a 2 × 1 matrix and adhere to probability norms.
Post 2: Reply to a Classmate
Assume your team member has passed the diagram and initial setup on to you for further analysis of how the likelihood of these outcomes may affect the company in the short term. Review a classmate’s state-transition diagram and address all of the following items.
- Translate the probabilities given in the diagram to a transition matrix, P. Present this as a 2 × 2 matrix.
- Generate the proportion in State A versus State B after the first time period (i.e., the distribution after one step). This is also known as the state matrix after one period, S1, found by multiplying the initial distribution vector, v, given by your classmate by the transition matrix, P, you derived from their diagram (S1 = v∙P). Your result for S1 should be expressed as a matrix/vector with the appropriate dimensions.
- Interpret the state matrix after one period, S1, which you just computed, in your own words, using one or two sentences. In another one or two sentences, express to your team member some ideas on how the company might find this information useful.
Post 3: Reply to Another Classmate
Assume your team members have now shared with you all of the information examined to date, including the diagram, initial setup, and analysis of what would be likely after one time period. Review a different classmate’s state-transition diagram and address all of the following items.
- Generate the proportions in State A versus State B for each period at least 20 periods from now, by selecting a value for n ≥ 20. Compute S2 and all subsequent distribution vectors for n steps into the future, by using the information presented by your classmates along with a computational tool of your choice (e.g., Excel®, Python®). Organize and present your results clearly as a summary table, showing the results for the distribution vectors for the first three periods and then also for the final three periods leading up to your nth state matrix/vector.
- What proportions are likely to be in State A and State B, long term? Address this question by interpreting the long-term trend you observe in the state matrices which you just computed, using one or two sentences. In another one or two sentences, express to your team members some ideas on how the company might find this information useful, relating it to your classmate’s original context.