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What is the definition of transition probabilities in a Markov Chain model?
Transition probabilities are the probabilities of moving from one state to another in one time step.
How are transition probabilities after one time step denoted?
The transition probability from state i to state j after one time step is denoted as $p_{ij}(1)$.
Define transition probabilities after n-step time period.
Transition probabilities after n-step time period are denoted as $p_{ij}(n)$.
How are initial probabilities defined in a Markov Chain model?
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What is the significance of Markov chains in stochastic processes?
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Why are transition probabilities crucial in analyzing Markov Chain models?
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How do you calculate the probability of being in state X2 on the 1st day?
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What does the Transition Probability Matrix (TPM) indicate?
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What is the purpose of the Initial State distribution matrix?
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How is the probability after n steps calculated in a Markov Chain?
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What does the sum of probabilities in each row of the Transition Probability Matrix (TPM) represent?
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How is the first state matrix obtained in a Markov Chain?
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What is a stochastic process?
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Differentiate between a stochastic process and a time series.
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Provide examples of stochastic processes.
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In a market with 3 mobile network operator companies (A, B, C), if a consumer is currently using company A's SIM, what is the probability that they will continue with the same company next year?
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If a consumer is currently using company A's SIM, what is the probability that they will shift to company B next year?
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If a consumer is currently using company A's SIM, what is the probability that they will shift to company C next year?
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Study Notes
Markov Chain Basics
- A Markov Chain is a mathematical model that describes a sequence of random events, where the probability of transitioning from one state to another is dependent on the current state.
Transition Probabilities
- Transition probabilities define the probability of transitioning from one state to another in a Markov Chain model.
- Transition probabilities after one time step are denoted as P(i, j), where P(i, j) represents the probability of transitioning from state i to state j.
- Transition probabilities after an n-step time period can be calculated by multiplying the one-step transition probabilities n times.
Initial Probabilities
- Initial probabilities define the probability distribution of the initial state in a Markov Chain model.
- The initial state distribution matrix represents the probability of being in each state at the initial time step.
Significance of Markov Chains
- Markov Chains are crucial in stochastic processes, as they provide a mathematical framework for modeling and analyzing complex systems that undergo random transitions between states.
Importance of Transition Probabilities
- Transition probabilities are crucial in analyzing Markov Chain models, as they enable the calculation of the probability of being in a particular state at a given time step.
Calculating Probabilities
- The probability of being in state X2 on the 1st day can be calculated using the transition probability matrix and the initial state distribution.
- The probability after n steps can be calculated by multiplying the one-step transition probabilities n times.
Transition Probability Matrix (TPM)
- The TPM indicates the probability of transitioning from one state to another in a single time step.
- The sum of probabilities in each row of the TPM represents the probability of being in any state after a single time step, which is always equal to 1.
Initial State Distribution Matrix
- The purpose of the Initial State distribution matrix is to represent the probability distribution of the initial state in a Markov Chain model.
Calculating First State Matrix
- The first state matrix can be obtained by multiplying the initial state distribution matrix with the transition probability matrix.
Stochastic Processes
- A stochastic process is a mathematical model that describes a sequence of random events, where the probability of each event is dependent on the previous events.
- A stochastic process can be stationary or non-stationary, and it can be discrete or continuous.
Differentiating Stochastic Processes and Time Series
- A stochastic process is a mathematical model that describes a sequence of random events, whereas a time series is a sequence of data points measured at regular time intervals.
Examples of Stochastic Processes
- Examples of stochastic processes include stock prices, weather patterns, and customer behavior.
Markov Chain Applications
- Markov Chain models can be applied to a wide range of real-world scenarios, such as predicting customer behavior, analyzing supply chains, and modeling population dynamics.
- In a market with 3 mobile network operator companies (A, B, C), if a consumer is currently using company A's SIM, the probability of continuing with the same company next year can be calculated using the transition probability matrix.
- The probability of shifting to company B or C next year can also be calculated using the transition probability matrix.
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Description
This quiz covers the calculation of probabilities in Markov Chains, including finding the probability of a specific state at a given time period, conditional probabilities, and multiple state probabilities. It also includes notations like q0 for initial probabilities, q1 for probabilities after 1 time period, and P for transition probability matrix.
