About State vs Sample
A state is a possible value the random variable can take — it exists whether or not you’ve observed it.
A sample is a realized observation from a distribution — a concrete outcome you observe.
Example
In a Markov chain modeling whether it rains or not:
- State space:
- Sample: “Today it rained” (a realized observation)
The state space exists theoretically as all possible outcomes. A sample is what you actually observe in reality.