todennäköisyysmatriisit

todennäköisyysmatriisit is a Finnish term that translates to "probability matrix" or "stochastic matrix" in English. It is a concept used in probability theory and linear algebra, particularly in the study of Markov chains.

A probability matrix is a square matrix where each entry represents the probability of transitioning from one state to another. The rows of the matrix represent the starting states, and the columns represent the ending states. The sum of probabilities in each row must equal 1, indicating that from any given state, there is a certainty of transitioning to some state (including staying in the same state). All entries in a probability matrix are non-negative and less than or equal to 1.

These matrices are fundamental to understanding how systems evolve over time. For example, in a Markov chain, the probability matrix describes the likelihood of moving between different states in a single step. By raising the probability matrix to a power, one can determine the probabilities of transitioning between states after multiple steps. This allows for the analysis of long-term behavior, such as convergence to a stationary distribution where the probabilities of being in each state no longer change. Applications of probability matrices are found in diverse fields, including finance, biology, physics, and computer science.