What is the initial state vector $S_0$ in this example?

Conditions

• The system has two states, A and B.
• The components are ordered consistently with the states, here A then B.
• The entries are probabilities describing the initial distribution.

Reasoning, step by step

1. Identify the states of the system, which are A and B.
2. Write the initial state vector $S_0$ as a column vector.
3. Assign the top component to state A and the bottom component to state B.
4. Interpret the numerical entries as the probabilities of starting in each respective state.

Example

The formula appears as $S_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. The speaker says the top component refers to state A and that 1 means a 100% probability it starts in state A, while the bottom component refers to state B and 0 means a 0% probability it starts in state B.

Common misconceptions

• Thinking that $S_0$ is derived from the transition probabilities rather than being an asserted initial condition.
• Assuming that a Markov-chain state vector always has only two entries, whereas the number of components equals the number of states.

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Explore next

• Which state does each component of $S_0$ correspond to?
• Is $S_0 = (1,0)^T$ the only possible initial state for this Markov chain?