Todennäköisyysvektorin

todennäköisyysvektori, known in English as a probability vector, is a vector whose elements are non-negative real numbers and sum to one. These vectors are fundamental in probability theory and statistics, particularly in the study of discrete probability distributions. Each element of a probability vector represents the probability of a specific outcome or event occurring.

For example, if a random variable can take on three possible values, the corresponding probability vector would have three elements, say $p_1, p_2, p_3$, such that $p_1 \geq 0$, $p_2 \geq 0$, $p_3 \geq 0$, and $p_1 + p_2 + p_3 = 1$. Here, $p_1$ would be the probability of the first outcome, $p_2$ the probability of the second, and $p_3$ the probability of the third.

Probability vectors are used to describe the state of a system in various probabilistic models. A common application is in Markov chains, where the probability vector represents the distribution of states at a particular time step. The transition matrix of a Markov chain, when applied to a probability vector, results in a new probability vector representing the distribution of states at the next time step. The property that the elements sum to one ensures that the probabilities remain normalized throughout the process.