In: Statistics and Probability
Suppose that the traffic on your way to work depends on the traffic the day before. There are 3 types of traffic: light, moderate, and heavy. If there is light traffic on day n, then there is a 30% chance there will be light traffic on the following day, and a 50% chance of having moderate traffic the following day. If there is moderate traffic on day n, then there is a 35% chance of having light traffic the next day, and a 25% chance of having moderate traffic the next day. If there is heavy traffic on day n, then there is a 50% chance the traffic will be light the next day and a 40% chance the traffic will be moderate. 3. a) (5 points) Construct the Markov chain. Please define (1) the states, (2) the one step transition matrix P, and (3) draw a visual representation of the Markov chain b) (4 points) Assume you will be late to work if the traffic is heavy (you make it on time if the traffic is either light or moderate). If you are late to work, then you are also late to your morning meeting, and you have a morning meeting Tuesday through Friday. Given that you are not late to your meeting on Tuesday, what is the probability that you are not late to your meeting on Friday? Please only write the specific equation to determine this value.
(a) The markov chain is as follows:
(1) The state space of the given Markov process is as follows:
State 1: LIGHT- The traffic on the way to work is light.
State 2: MODERATE- The traffic on the way to work is moderate.
State 1: HEAVY- The traffic on the way to work is heavy.
(2) The one step transition matrix P is given as follows:
LIGHT | MODERATE | HEAVY | |
LIGHT | 0.30 | 0.50 | 0.20 |
MODERATE | 0.35 | 0.25 | 0.40 |
HEAVY | 0.50 | 0.40 | 0.10 |
Note that you can calculate the missing probabilities by using the property that the sum of each row in a transition matrix is 1. That is atleast transition to one of the states will happen or the process will stay in the same state.
(3) The transition diagram is given as follows:
Tuesday to Friday been there will be 3 tansitions. To calculate the required probability we need to calculate the P3 matrix, which is given as:
LIGHT | MODERATE | HEAVY | |
LIGHT | 0.37375 | 0.38325 | 0.243 |
MODERATE | 0.361875 | 0.379625 | 0.2585 |
HEAVY | 0.3735 | 0.3755 | 0.251 |
So, the total probability that the traffic will be light or moderate on Friday given that the traffic was light or moderate on Tuesday is 1-0.243-0.2585=0.4985.