In a large metropolitan area, 25% of the commuters currently use the public transportation system, whereas the remaining 75% commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now 30% of those who are now commuting to work via automobile will switch to public transportation, and 70% will continue to commute via automobile. At the same time, it is expected that 20% of those now using public transportation will commute via automobile and 80% will continue to use public transportation.
(a) Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters.
P A | ||||||||||||||||||
T | = |
|
|
(b) Find the initial distribution vector for this Markov chain.
X0 |
= |
|
(c) What percentage of the commuters are expected to use public
transportation 6 months from now? (Round your answer to the nearest
percent.)
%
ANSWER:
Let the states of the Markov chain be P and A denoting public transportation and automobile respectively.
The transition probability from P to P and A are 0.8 and 0.2 respectively.
The transition probability from A to P and A are 0.3 and 0.7 respectively.
a)
The transition matrix for the Markov chain is,
transportation used by these commuters.
P A | ||||||
T | = |
|
0.8 0.2 0.3 0.7 |
b)
Given, 25% of the commuters currently use the public transportation system, whereas the remaining 75% commute via automobile.
So, the initial distribution vector for this Markov chain is
X0 = [ 0.25 , 0.75]
c)
State after 6 months
X1 = X0 P
= [ 0.25 0.75]
= [0.25 * 0.8 + 0.75 * 0.3 0.25 * 0.2 + 0.75 * 0.7]
= [0.425 0.575]
Percentage of the commuters are expected to use public transportation 6 months from now = 0.425 = 42.5%.
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In a large metropolitan area, 25% of the commuters currently use the public transportation system, whereas...
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