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Q2. People often decide their outdoor activities according to the weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by W = (W1, W2, W3), is unknown. His activities option, denoted by V = (v₁, V2, V3), is decided by the weather conditions. The initial state of the weather is 7 = [0.3, 0.4, 0.3]. Given the Hidden Markov model 0 = (A, B, T), calculate the probability that you observe a specific activity sequence O [V2, V2, V1, V3] of your friend over the past four days, where A₁, is the transition probability from wi to wj, Bij is the probability of observing the activity v, under the state wi. = A = [0.3 0.2 0.5] 0.1 0.4 0.5 B 0.2 0.5 0.3 = [0.4 0.5 0.1] 0.2 0.4 0.4 0.3 0.1 0.6

Q2. People often decide their outdoor activities according to the weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by W = (W1, W2, W3), is unknown. His activities option, denoted by V = (v₁, V2, V3), is decided by the weather conditions. The initial state of the weather is 7 = [0.3, 0.4, 0.3]. Given the Hidden Markov model 0 = (A, B, T), calculate the probability that you observe a specific activity sequence O [V2, V2, V1, V3] of your friend over the past four days, where A₁, is the transition probability from wi to wj, Bij is the probability of observing the activity v, under the state wi. = A = [0.3 0.2 0.5] 0.1 0.4 0.5 B 0.2 0.5 0.3 = [0.4 0.5 0.1] 0.2 0.4 0.4 0.3 0.1 0.6

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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**Question 2.** People often decide their outdoor activities according to the weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by \( W = (\omega_1, \omega_2, \omega_3) \), is unknown. His activities option, denoted by \( V = (v_1, v_2, v_3) \), is decided by the weather conditions. The initial state of the weather is \( \pi = [0.3, 0.4, 0.3] \). Given the Hidden Markov model \( \theta = (A, B, \pi) \), calculate the probability that you observe a specific activity sequence \( O = [v_2, v_2, v_1, v_3] \) of your friend over the past four days, where \( A_{i,j} \) is the transition probability from \( \omega_i \) to \( \omega_j \), \( B_{i,j} \) is the probability of observing the activity \( v_j \) under the state \( \omega_i \). The transition matrix \( A \) and the observation matrix \( B \) are given as follows: \[ A = \begin{bmatrix} 0.3 & 0.2 & 0.5 \\ 0.1 & 0.4 & 0.5 \\ 0.2 & 0.5 & 0.3 \\ \end{bmatrix} \] \[ B = \begin{bmatrix} 0.4 & 0.5 & 0.1 \\ 0.2 & 0.4 & 0.4 \\ 0.3 & 0.1 & 0.6 \\ \end{bmatrix} \]
Transcribed Image Text:**Question 2.** People often decide their outdoor activities according to the weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by \( W = (\omega_1, \omega_2, \omega_3) \), is unknown. His activities option, denoted by \( V = (v_1, v_2, v_3) \), is decided by the weather conditions. The initial state of the weather is \( \pi = [0.3, 0.4, 0.3] \). Given the Hidden Markov model \( \theta = (A, B, \pi) \), calculate the probability that you observe a specific activity sequence \( O = [v_2, v_2, v_1, v_3] \) of your friend over the past four days, where \( A_{i,j} \) is the transition probability from \( \omega_i \) to \( \omega_j \), \( B_{i,j} \) is the probability of observing the activity \( v_j \) under the state \( \omega_i \). The transition matrix \( A \) and the observation matrix \( B \) are given as follows: \[ A = \begin{bmatrix} 0.3 & 0.2 & 0.5 \\ 0.1 & 0.4 & 0.5 \\ 0.2 & 0.5 & 0.3 \\ \end{bmatrix} \] \[ B = \begin{bmatrix} 0.4 & 0.5 & 0.1 \\ 0.2 & 0.4 & 0.4 \\ 0.3 & 0.1 & 0.6 \\ \end{bmatrix} \]
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