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What does it mean for states to be mutually exclusive?
Which statement best defines exhaustive states?
If two events are mutually exclusive, what can be inferred about their occurrence?
Which scenario illustrates the concept of exhaustive states?
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In the context of states, which of the following best highlights the difference between mutual exclusivity and exhaustive inclusion?
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What must be true about all entries in the transition probability matrix P?
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What does the equation $\sum_{j=0}^{m} P_{ij} = 1$ signify in the context of a transition probability matrix?
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In the transition probability matrix, what does the subscript notation P_{ij} represent?
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What is the implication of having $P_{ij} < 0$ for any i, j in the transition probability matrix?
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What is the significance of the symbol $\sigma_m$ in relation to the transition probability matrix?
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What type of values can a stochastic process take on?
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Which of the following accurately describes the range of a stochastic process as defined in the content?
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In the context of a stochastic process, what does the notation {𝑋𝑡 , t=0, 1, 2, 3, …} indicate?
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Why are the values (states) of the stochastic process considered non-negative?
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What is the maximum value represented in the finite set of state values for the stochastic process?
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What represents the initial probability distribution in a Markov chain?
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If $P_{Xo} = 2$, which term in the vector q does this correspond to?
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Which of the following statements is true regarding the probabilities $q_0$, $q_1$, $q_2$, and $q_3$?
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Given the notation, how many distinct probabilities are represented in the vector q if it is defined as $q = [q_0, q_1, q_2, ext{...}, q_m]$?
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In the context of probability distributions for Markov chains, what is the primary purpose of the vector q?
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What does 𝑋𝑡 represent in the context provided?
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If 𝑆 is set to 3, how many states are there in total?
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What is the possible range of states in this policy when 𝑆 = 3?
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What is represented by the expression 𝑋𝑡+1?
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What does the variable 𝑆 signify in this context?
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Study Notes
Stochastic Processes and Markov Chains
- A stochastic process is represented as {𝑋𝑡, t=0, 1, 2, 3, …}, which can take on a finite or countable number of states.
- The states of the process are denoted by non-negative integers {0, 1, 2, …, m}, representing mutually exclusive and exhaustive outcomes.
- Mutually Exclusive: States cannot overlap; a process cannot be in more than one state simultaneously.
- Exhaustive: All possible states are covered, ensuring complete outcomes.
Initial Probability Distribution
- The initial probability distribution is defined as:
- 𝑃 𝑋𝑜 = 0 = 𝑞0
- 𝑃 𝑋𝑜 = 1 = 𝑞1
- 𝑃 𝑋𝑜 = 2 = 𝑞2
- 𝑃 𝑋𝑜 = 3 = 𝑞3
- This distribution can be expressed as a vector: q = [𝑞0, 𝑞1, 𝑞2, …, 𝑞𝑚].
Transition Probability Matrix
- In a Markov chain, the transition probability matrix is crucial:
- The sum of probabilities from one state to the next must equal 1:
- σ𝑗=0 𝑃 𝑋𝑡+1 = 𝑗 | 𝑥𝑡 = 𝑖 = 1
- Each entry in the matrix must be non-negative:
- 𝑃𝑖𝑗 ≥ 0 for all pairs of states i and j.
- The sum of probabilities from one state to the next must equal 1:
Example Application
- Consider time measured in weeks with states {0, 1, 2, ..., S}.
- Let 𝑋0 represent the initial number of cameras available, defining the process’s starting condition.
Key Properties of Markov Chains
- Future states depend solely on the current state, not on the sequence of events that preceded it.
- Establishing initial probabilities and transitions is fundamental for modeling and predicting the behavior of the stochastic process.
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Description
This quiz covers the fundamentals of stochastic processes and Markov chains, focusing on key concepts such as initial probability distribution and transition probability matrices. Understand mutually exclusive states and the mathematical formulation of these processes. Test your knowledge on how probabilities transition between different states in a Markov chain.
