Final answer:
To construct a two-state transition probability matrix for a Markov chain with given probabilities, fill in the matrix with the given probabilities and use the complementary probability principle for the remaining probabilities. The result is a matrix where each row sums to 1, showing the transitions between a dry day and a rainy day.
Step-by-step explanation:
The student is asking about how to construct a two-state transition probability matrix for a Markov chain in which there are two possible weather states: a dry day (state 0) and a rainy day (state 1). Given that the probability of a dry day following a rainy day is 1/3, and the probability of a rainy day following a dry day is 1/2, we can infer the remaining probabilities by using the complementary probability principle since the sum of probabilities for the transitions from each state must equal 1.
The two transition probabilities we have are:
• Probability of a dry day following a rainy day (P(0|1)) = 1/3
• Probability of a rainy day following a dry day (P(1|0)) = 1/2
Therefore:
• Probability of a rainy day following a rainy day (P(1|1)) = 1 - P(0|1) = 1 - 1/3 = 2/3
• Probability of a dry day following a dry day (P(0|0)) = 1 - P(1|0) = 1 - 1/2 = 1/2
With this, we can now construct the transition probability matrix for the Markov chain:
State 0State 1From State 01/21/2From State 11/32/3
In matrix form, this transition probability matrix can be represented as:
[ [1/2, 1/2], [1/3, 2/3] ]