Siirtymätodennäköisyyksiksi

Siirtymätodennäköisyyksiksi, also known as transition probabilities, are fundamental concepts in probability theory and stochastic processes, particularly in the context of Markov chains. These probabilities quantify the likelihood of moving from one state to another within a defined system over a single time step. In essence, they describe the dynamics of a system that transitions between different states, where the future state depends only on the current state and not on the sequence of events that preceded it.

For a system with a finite number of states, say $S = \{s_1, s_2, ..., s_n\}$, the transition probabilities can be represented by a transition matrix. This matrix, often denoted by $P$, is a square matrix where each element $p_{ij}$ represents the probability of transitioning from state $s_i$ to state $s_j$. The rows of the transition matrix sum to 1, indicating that from any given state, there is always a transition to some state, including possibly remaining in the same state. The interpretation of these probabilities is crucial for understanding the long-term behavior and characteristics of the system being modeled.