Question 1:

Suppose we have three firms: a manufacturer, a distributor, and a retailer. At each firm i, let Xi be a binary random variable that represents if a supply chain disruption has occurred at firm i. That is, Xm will equal 1 if a disruption occurs at the manufacturer, and will equal 0 if a disruption does not occur at the manufacturer. Likewise, Xd will equal 1 if a disruption occurs at the distributor, and will equal 0 if a disruption does not occur at the distributor, and Xr will equal 1 if a disruption occurs at the retailer, and will equal 0 if a disruption does not occur at the retailer. Suppose that the probability of a disruption at the manufacturer is 0.1. Further suppose that the probability of disruption at the distributor depends on whether or not there is a disruption at the manufacturer. If there is no disruption at the manufacturer, then the probability of disruption at the distributor is 0.2. If there is a disruption at the manufacturer, then the probability of a disruption at the distributor is 0.6. Likewise, the probability of disruption at the retailer is dependent on what happens at the distributor. If there is no disruption at the distributor, then the probability of disruption at the retailer is 0.6. If there is a disruption at the distributor, then the probability of disruption at the retailer is 0.8. Using this information, find the following:

(a) The probability that a disruption will occur at the distributor given no other informa- tion about disruptions elsewhere.

(b) The probability that a disruption will occur at the retailer given no other information about disruptions elsewhere.

(c) The probability that a disruption will occur at the retailer given that a disruption occurred at the manufacturer.

(d) The probability of disruption at the manufacturer given that we know there was a disruption at the retailer (Hint, use Bayes Theorem!)

(e) The expected number of locations to experience a disruption.

(f) The variance of the number of locations to experience a disruption.

(g) The expected number of locations to experience a disruption given that we know a disruption occurred at the distributor (do not count the distributor in this calculation)

(h) The variance of the number of locations to experience a disruption given that we know a disruption occurred at the distributor (again, do not count the distributor in this calculation)

Question 2:

Suppose that the movement of a stock price depends on the movement of the stock price the day before, along with the movement of the overall market the day before. In other words, we have a system where the stock prices can either move up, down, or remain the same. Likewise, the overall market can move up, down, or remain the same. Model this system as a Markov Chain. Think carefully about how you would model the states. Assume we are working in discrete time, where time is a single day. Once you model it, pick a stock and a market indicator (for example, Apple and the S&P 500). Obtain daily stock data for all of 2019 started at the first trading day of the year. Convert this stock data to data on where the system was in your modeled states. The data for the stock and the market should be two variables with acceptable values of “UP”, “DOWN”, or “UNCHANGED”. Compute the conditional probabilities of moving from one of your states to another. Last, compute the steady-state probability distribution using matrix algebra (you are allowed to use a computer to solve the system, but you are required to write the system and interpret it).

Attachments: