Exam 12: Markov Analysis

Judy Jones purchases groceries and pop exactly once each week on Sunday evenings. She buys either Coke or Pepsi only and switches from Coke to Pepsi and vice-versa somewhat regularly. Her purchasing behavior of these two drinks is modeled as a Markov system. Let the transition matrix be: Let $\mathrm{P}($ Coke $)$ and $\mathrm{P}(\mathrm{Pepsi})$ respectively denote the steady state probability that Judy will buy Coke or Pepsi in the very long run on any week. Which of the following is the correct system of equations to find these steady state probabilities?

Newsweek and Time are two competing weeklies, each of which tries to keep their readership while trying to get the other's readers to switch. Among all the households holding yearly subscription to Newsweek or Time but not both, let $\mathrm{N}$ and $\mathrm{T}$ denote the states that a household is a current Newsweek or Time subscriber, respectively. The probabilities of switching from one state to the other after one transition (when they renew) are given by the tree diagram. (For simplicity, it may be assumed that all annual subscriptions expire on December 31st of the year and are renewed by all of them for one or the other but not both magazines for one year.) If the current number of households subscribing for Newsweek and Time is, respectively 2000 and 3000, what would be the number of subscribers to Newsweek after one renewal?

XYZ Inc. hires only retired people for its greeter's job. They always start on January 1 and are considered for promotion every January 1st thereafter. On every January 1st, any existing greeter (state G) - those who were greeters last year - may be retired (state R), may be dead (sate D), may continue as a greeter (state G), or may be promoted as greeter -in-chief (state GIC). Persons in state GIC may continue in it or move to states $\mathrm{D}$ or $\mathrm{R}$ in one year. The following is the transition matrix. This matrix has a set of cyclical states, namely states 3 and 4 .

In Markov systems, the probability of going from one state in period $n$ to another state in period $(n+1)$ depends on what states the system traveled in periods $1,2, \ldots$ , n.

If there are only two absorbing states and three transient states, we can add a dummy absorbing state to make the matrix I-Q into a square matrix.

Judy Jones purchases groceries and pop exactly once each week on Sunday evenings. She buys either Coke or Pepsi only and switches from Coke to Pepsi and vice-versa somewhat regularly. Her purchasing behavior of these two drinks is modeled as a Markov system. Let the transition matrix be: In the next 700 weeks, on an average, how many weeks is Judy expected to purchase Pepsi?

XYZ Inc. hires only retired people for its greeter's job. They always start on January 1 and are considered for promotion every January 1st thereafter. On every January 1st, any existing greeter (state G) - those who were greeters last year - may be retired (state R), may be dead (sate D), may continue as a greeter (state G), or may be promoted as greeter -in-chief (state GIC). Persons in state GIC may continue in it or move to states $\mathrm{D}$ or $\mathrm{R}$ in one year. The following is the transition matrix. This matrix has three transient states.

A state or a group of states in a Markov chain is called absorbing if the system can never get out of that state or the group of states (if it ever reaches it).

A study was made about the incidence of hard-working fathers having lazy sons and lazy fathers having hard-working sons among several generations of families having only one son. It was hypothesized (based on a preliminary study) that a Markov chain model described below will fit the profile from one generation to the next. If in a particular generation the father was hard-working, what is the probability that his son would be lazy?

Jim Cramer, a stock analyst, models the movement of the closing price in NYSE of the stock WidgetsR-Us Inc. (symbol: WRU) as a Markov chain with a transition time of 1 day. There are two states in this Markov system-State A, the closing price increased or stayed the same from the previous day and State $\mathrm{B}$ , the closing price decreased from the previous day. Suppose that the system is on State A at the end of today, what is the probability that it will be in State A after two trading days (48 hours)?

Judy Jones purchases groceries and pop exactly once each week on Sunday evenings. She buys either Coke or Pepsi only and switches from Coke to Pepsi and vice-versa somewhat regularly. Her purchasing behavior of these two drinks is modeled as a Markov system. Let the transition matrix be: Let $\mathrm{P}($ Coke) and $\mathrm{P}(\mathrm{Pepsi})$ respectively denote the steady state probability that Judy will buy Coke or Pepsi in the very long run on any week. P(Coke) for this data will be

Newsweek and Time are two competing weeklies, each of which tries to keep their readership while trying to get the other's readers to switch. Among all the households holding yearly subscription to Newsweek or Time but not both, let $\mathrm{N}$ and $\mathrm{T}$ denote the states that a household is a current Newsweek or Time subscriber, respectively. The probabilities of switching from one state to the other after one transition (when they renew) are given by the tree diagram. (For simplicity, it may be assumed that all annual subscriptions expire on December 31st of the year and are renewed by all of them for one or the other but not both magazines for one year.) If the current number of households subscribing for Newsweek and Time is, respectively, 2000 and 3000, what would be the number of subscribers to Time after two renewals?

XYZ Inc. hires only retired people for its greeter's job. They always start on January 1 and are considered for promotion every January 1st thereafter. On every January 1st, any existing greeter (state G) those who were greeters last year - may be retired (state R), may be dead (sate D), may continue as a greeter (state G), or may be promoted as greeter -in-chief (state GIC). Persons in state GIC may continue in it or move to states D or R in one year. The following is the transition matrix. While trying to find the probability of eventual absorption in the absorbing states, Matrix $\mathrm{Q}$ for this problem is:

Newsweek and Time are two competing, weeklies each of which tries to keep their readership while at the same time trying to get the other's readers to switch. Among all the households holding yearly subscriptions to Newsweek or Time (but not both), let $\mathrm{N}$ and $\mathrm{T}$ denote the states that a household is a current Newsweek or Time subscriber, respectively. The probabilities of switching from one state to the other after one transition (when they renew) are given by the tree diagram. (For simplicity, it may be assumed that all annual subscriptions expire on December 31st and are renewed by all of them for one or the other but not both magazines for one year.) If the current number of households subscribing to Newsweek and Time is 2000 and 4000, respectively, Time would have a larger number of subscribers in the long run.

Which of the following is not a part of the characteristics of a Markov system?

Newsweek and Time are two competing weeklies, each of which tries to keep their readership while trying to get the other's readers to switch. Among all the households holding yearly subscription to Newsweek or Time but not both, let $\mathrm{N}$ and $\mathrm{T}$ denote the states that a household is a current Newsweek or Time subscriber, respectively. The probabilities of switching from one state to the other after one transition (when they renew) are given by the tree diagram. (For simplicity, it may be assumed that all annual subscriptions expire on December 31st of the year and are renewed by all of them for one or the other but not both magazines for one year.) If the current number of households subscribing for Newsweek and Time respectively was 4000 and 2000, then both magazines would have the same number of subscribers in the very long run.

Inverse of a matrix exists even if the matrix in not a square matrix.

XYZ Inc. hires only retired people for its greeter's job. They always start on January 1 and are considered for promotion every January 1st thereafter. On every January 1st, any existing greeter (state G) those who were greeters last year - may be retired (state R), may be dead (sate D), may continue as a greeter (state G), or may be promoted as greeter -in-chief (state GIC). Persons in state GIC may continue in it or move to states $\mathrm{D}$ or $\mathrm{R}$ in one year. The following is the transition matrix. While trying to find the probability of eventual absorption in the absorbing states, Matrix $\mathrm{N}=(\mathrm{I}-\mathrm{Q})$ for this problem is:

Joe Smith, a loyal lessee of American cars, changes cars exactly once every two years. Joe leases either a Ford or a Chrysler only. His leasing behavior is modeled as a Markov chain, and the transition matrix is given below. Let $\mathrm{P}$ (Ford) and $\mathrm{P}($ Chrysler) respectively denote the steady state probability that Joe will lease a Ford or a Chrysler in the long run. P(Ford) for this data will be

Vikram eats a sandwich for dinner every Sunday. He only likes Papa John's or Potbelly. His behavior in this context is captured by the following Markov transition matrix. Suppose on the first Sunday of 2006, Vikram ate in Potbelly, what is the probability that he would eat Potbelly's on the second Sunday of 2006?