Search results
Results from the WOW.Com Content Network
Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs and position of the runners.
A Markov chain is a stochastic process defined by a set of states and, for each state, a probability distribution on the states. Starting from an initial state, it follows a sequence of states where each state in the sequence is chosen randomly from the distribution associated with the previous state.
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and = holds, then by summing over , the global balance equations are satisfied and is the stationary distribution of the process. [5]
Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,
Due to the secret passageway, the Markov chain is also aperiodic, because the ghosts can move from any state to any state both in an even and in an uneven number of state transitions. Therefore, a unique stationary distribution exists and can be found by solving π Q = 0 {\displaystyle \pi Q=0} , subject to the constraint that elements must sum ...
In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution.
Markov chains with generator matrices or block matrices of this form are called M/G/1 type Markov chains, [13] a term coined by Marcel F. Neuts. [ 14 ] [ 15 ] An M/G/1 queue has a stationary distribution if and only if the traffic intensity ρ = λ E ( G ) {\displaystyle \rho =\lambda \mathbb {E} (G)} is less than 1, in which case the unique ...