siirtymätodennäköisyysmatriisia

siirtymätodennäköisyysmatriisia, often translated as transition probability matrix or simply transition matrix, is a fundamental concept in the study of Markov chains. It is a square matrix where each element P(i, j) represents the probability of transitioning from state i to state j in a single step. The rows of the matrix sum to 1, reflecting that from any given state, there must be a transition to some state (including possibly staying in the same state). The columns also sum to 1, but this property is not as immediately intuitive as the row sum property.

The dimensions of the matrix correspond to the number of possible states in the system. If a Markov chain has n states, the siirtymätodennäköisyysmatriisia will be an n x n matrix. This matrix allows for the prediction of future states of the system based on its current state and the defined probabilities. Multiplying the transition matrix by a state vector (a vector representing the probability distribution across states) yields the state vector for the next time step. Repeated multiplication of the transition matrix by itself, i.e., calculating powers of the matrix, can reveal long-term behavior, such as steady-state distributions.