Math Problem Statement
Solution
The example discusses a Markov chain where a fair die is tossed repeatedly, and denotes the maximum number observed in the first tosses. The solution requires finding:
- The transition probability matrix ,
- , and
- .
Here’s a breakdown of the key concepts:
State Space:
The state space is defined as , representing the possible maximum values that can occur in the first tosses.
Transition Rules:
For a given maximum , the maximum depends on the next roll:
- , if the next roll .
- (), if the next roll equals .
Transition Probabilities:
For a specific :
- The probability that : This happens if the next roll is one of the numbers , which is .
- The probability that (): This happens if the next roll is , which is .
Thus, the entries in the transition probability matrix are:
\begin{cases} \frac{i}{6}, & \text{if } j = i, \\ \frac{1}{6}, & \text{if } j > i, \\ 0, & \text{if } j < i. \end{cases}$$ For example: - For $$X_n = 3$$, we have: - $$P(X_{n+1} = 3 \mid X_n = 3) = \frac{3}{6}$$, - $$P(X_{n+1} = 4 \mid X_n = 3) = \frac{1}{6}$$, - $$P(X_{n+1} = 5 \mid X_n = 3) = \frac{1}{6}$$, - $$P(X_{n+1} = 6 \mid X_n = 3) = \frac{1}{6}$$. --- ### **Finding $$P^2$$**: To compute $$P^2$$, multiply the transition matrix $$P$$ by itself. Each entry $$(i, j)$$ in $$P^2$$ represents the probability of transitioning from state $$i$$ to state $$j$$ in two steps. --- ### **Finding $$P(X_2 = 6)$$**: $$P(X_2 = 6)$$ is the probability that the maximum of the first two rolls is 6. This is computed using the transition probabilities from the initial state. --- Would you like me to: 1. Explicitly write the transition matrix $$P$$? 2. Compute $$P^2$$ in detail? 3. Derive $$P(X_2 = 6)$$? 4. Provide a general explanation of Markov chains? 5. Explore another related concept? **Tip:** Always identify the state space and transition rules clearly in any Markov chain problem!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Markov Chains
Probability
Transition Matrices
Formulas
P(i, j) = i/6 if j = i
P(i, j) = 1/6 if j > i
P(i, j) = 0 if j < i
Theorems
Markov Property
Suitable Grade Level
Undergraduate
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