(i) If volume is high this week, then next week it will be high with a probability of 0.8 and low with a probability of 0.2. (ii) If volume is low this week then it will be high next week with a probability of 0.3. The manager estimates that the volume is three times as likely to be high as to be low this week. Assume that state 1 is high volume and that state 2 is low volume. (Note: Express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).) (1) Find the transition matrix P for this Markov chain: .8 2 P = .3 .7 (2) Find the state vector that represents the manager's estimate | 314 X, = 1/4 (3) Using this estimate as the initial state vector, find the state vector for two weeks from now: X2 = What is the probability that two weeks from now the volume will be high? (4) Again, using the manager's estimate as the initial state vector, find the state vector for three weeks from now: X3 = What is the probability that three weeks from now the volume will be high? (5) Suppose, contrary to the manager's estimate, that this week the volume is low. How many weeks must pass before a week comes along in which the probability of high volume is at least 0.5?
(i) If volume is high this week, then next week it will be high with a probability of 0.8 and low with a probability of 0.2. (ii) If volume is low this week then it will be high next week with a probability of 0.3. The manager estimates that the volume is three times as likely to be high as to be low this week. Assume that state 1 is high volume and that state 2 is low volume. (Note: Express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).) (1) Find the transition matrix P for this Markov chain: .8 2 P = .3 .7 (2) Find the state vector that represents the manager's estimate | 314 X, = 1/4 (3) Using this estimate as the initial state vector, find the state vector for two weeks from now: X2 = What is the probability that two weeks from now the volume will be high? (4) Again, using the manager's estimate as the initial state vector, find the state vector for three weeks from now: X3 = What is the probability that three weeks from now the volume will be high? (5) Suppose, contrary to the manager's estimate, that this week the volume is low. How many weeks must pass before a week comes along in which the probability of high volume is at least 0.5?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(i) If volume is high this week, then next week it will be high with a probability of 0.8 and low with a probability of 0.2.
(ii) If volume is low this week then it will be high next week with a probability of 0.3.
The manager estimates that the volume is three times as likely to be high as to be low this week.
Assume that state 1 is high volume and that state 2 is low volume.
(Note: Express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).)
(1) Find the transition matrix P for this Markov chain:
.8
.2
P =
.3
.7
(2) Find the state vector that represents the manager's estimate
Xo =
3/4
1/4
(3) Using this estimate as the initial state vector, find the state vector for two weeks from now:
X2 =|
What is the probability that two weeks from now the volume will be high?
(4) Again, using the manager's estimate as the initial state vector, find the state vector for three weeks from now:
X3 =
What is the probability that three weeks from now the volume will be high?
(5) Suppose, contrary to the manager's estimate, that this week the volume is low. How many weeks must pass before a week comes along in which the probability of high volume is
at least 0.5?
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