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Problem 4: A finite state machine has 3 states, labeled {1, 2, 3}. The input of this state machine is a binary value; it can only be 0 or 1. The probability of the input being 0 is 0.4. (a) Plot the state transition diagram and find the transition probability matrix given that: • When the input is 1 the machine moves to the state above the current state (2 is above 1, 3 is above 2, 1 is above 3). • When the input is 0 the machine moves to the state below the current state (1 is below 2, 2 is below 3, 3 is below 1). • These are the only two possible transitions. (b) If the process starts in state 1 (Xo = 1), what is the probability that it will be in state 2 after 3 transitions (X3 = 2)? (c) If we know that the process starts in state 3 (Xo = 3), what are the 3 state probabilities after two transitions (that is, at time k = 2)?

Problem 4: A finite state machine has 3 states, labeled {1, 2, 3}. The input of this state machine is a binary value; it can only be 0 or 1. The probability of the input being 0 is 0.4. (a) Plot the state transition diagram and find the transition probability matrix given that: • When the input is 1 the machine moves to the state above the current state (2 is above 1, 3 is above 2, 1 is above 3). • When the input is 0 the machine moves to the state below the current state (1 is below 2, 2 is below 3, 3 is below 1). • These are the only two possible transitions. (b) If the process starts in state 1 (Xo = 1), what is the probability that it will be in state 2 after 3 transitions (X3 = 2)? (c) If we know that the process starts in state 3 (Xo = 3), what are the 3 state probabilities after two transitions (that is, at time k = 2)?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Problem 4: A finite state machine has 3 states, labeled {1, 2, 3}. The input of this state machine is a binary value; it can only be 0 or 1. The probability of the input being 0 is 0.4. (a) Plot the state transition diagram and find the transition probability matrix given that: . When the input is 1 the machine moves to the state above the current state (2 is above 1, 3 is above 2, 1 is above 3). . When the input is 0 the machine moves to the state below the current state (1 is below 2, 2 is below 3, 3 is below 1). These are the only two possible transitions. (b) If the process starts in state 1 (Xo = 1), what is the probability that it will be in state 2 after 3 transitions (X3 = 2)? (c) If we know that the process starts in state 3 (Xo = 3), what are the 3 state probabilities after two transitions (that is, at time k = 2)?
Transcribed Image Text:Problem 4: A finite state machine has 3 states, labeled {1, 2, 3}. The input of this state machine is a binary value; it can only be 0 or 1. The probability of the input being 0 is 0.4. (a) Plot the state transition diagram and find the transition probability matrix given that: . When the input is 1 the machine moves to the state above the current state (2 is above 1, 3 is above 2, 1 is above 3). . When the input is 0 the machine moves to the state below the current state (1 is below 2, 2 is below 3, 3 is below 1). These are the only two possible transitions. (b) If the process starts in state 1 (Xo = 1), what is the probability that it will be in state 2 after 3 transitions (X3 = 2)? (c) If we know that the process starts in state 3 (Xo = 3), what are the 3 state probabilities after two transitions (that is, at time k = 2)?
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