Matrixes of transition arise by consideration of Markov chains which are a special case of Markov processes. The property defining them is that the condition of process in "future", depends on current state (in the present) and, at the same time, is not connected with "past".
Instruction
1. It is necessary to consider the accidental process (AP) of X (t). Its probabilistic description is based on consideration of n-dimensional density of probability of its sections W (x1, x2, …, xn; t1, t2, …, tn) which, on the basis of the device of conditional density of probability, can be rewritten in the form of W (x1, x2, …, xn; t1, t2, …, tn) = W (x1, x2, …, x (n-1); t1, t2, …, t(n-1)) by ∙W (xn, tn | x1, t1, x2, t2, …, x (n-1), t(n-1)), considering that t1
Definition. The joint venture, for which at any consecutive timepoints of t1
Using the device of the same conditional density of probabilities, it is possible to come to a conclusion, chtow (x1, x2, …, x (n-1), xn, tn; t1, t2, …, t(n-1), tn) = W (x1, tn) ∙W (x2, t2 | x1, t1) … ∙W (xn, tn | x (n-1), t(n-1)). Thus, all conditions of Markov process completely are defined by its initial condition and density of probabilities of transitions of W (xn, tn | the X (t(n-1)) =x(n-1))). For the discrete sequences (possible states and time are discrete) where instead of density of probabilities of transitions there are their probabilities and matrixes of transitions, the process carries the name - Markov's chain.
Consider a uniform chain of Markov (there is no dependence on time). Matrixes of transition are formed from conditional probabilities of transition of p(ij) (see fig. 1). It is the probability that for one step the system which had a state equal xi will pass into a condition of xj. Probabilities of transitions problem statement and its physical defines sense. Setting up them in a matrix receive the answer for this task.
Standard examples of creation of matrixes of transition give tasks about the wandering particles. Example. Let the system have five conditions of x1, x2, x3, x4, x5. The first and fifth are boundary. Let on each step the system can pass only into the state, next on number, and at the movement h5 about the probability of p, a h1 about the probability of q (p+q=1) aside aside. At achievement of borders the system can pass in h3 about the probability of v or remain in a former state with veroyatnostyyu1-v. Decision. In order that the task became absolutely transparent construct a state graph (see fig. 2).
2. Definition. The joint venture, for which at any consecutive timepoints of t1
Using the device of the same conditional density of probabilities, it is possible to come to a conclusion, chtow (x1, x2, …, x (n-1), xn, tn; t1, t2, …, t(n-1), tn) = W (x1, tn) ∙W (x2, t2 | x1, t1) … ∙W (xn, tn | x (n-1), t(n-1)). Thus, all conditions of Markov process completely are defined by its initial condition and density of probabilities of transitions of W (xn, tn | the X (t(n-1)) =x(n-1))). For the discrete sequences (possible states and time are discrete) where instead of density of probabilities of transitions there are their probabilities and matrixes of transitions, the process carries the name - Markov's chain.
Consider a uniform chain of Markov (there is no dependence on time). Matrixes of transition are formed from conditional probabilities of transition of p(ij) (see fig. 1). It is the probability that for one step the system which had a state equal xi will pass into a condition of xj. Probabilities of transitions problem statement and its physical defines sense. Setting up them in a matrix receive the answer for this task.
Standard examples of creation of matrixes of transition give tasks about the wandering particles. Example. Let the system have five conditions of x1, x2, x3, x4, x5. The first and fifth are boundary. Let on each step the system can pass only into the state, next on number, and at the movement h5 about the probability of p, a h1 about the probability of q (p+q=1) aside aside. At achievement of borders the system can pass in h3 about the probability of v or remain in a former state with veroyatnostyyu1-v. Decision. In order that the task became absolutely transparent construct a state graph (see fig. 2).
3. Using the device of the same conditional density of probabilities, it is possible to come to a conclusion, chtow (x1, x2, …, x (n-1), xn, tn; t1, t2, …, t(n-1), tn) = W (x1, tn) ∙W (x2, t2 | x1, t1) … ∙W (xn, tn | x (n-1), t(n-1)). Thus, all conditions of Markov process completely are defined by its initial condition and density of probabilities of transitions of W (xn, tn | the X (t(n-1)) =x(n-1))). For the discrete sequences (possible states and time are discrete) where instead of density of probabilities of transitions there are their probabilities and matrixes of transitions, the process carries the name - Markov's chain.
4. Consider a uniform chain of Markov (there is no dependence on time). Matrixes of transition are formed from conditional probabilities of transition of p(ij) (see fig. 1). It is the probability that for one step the system which had a state equal xi will pass into a condition of xj. Probabilities of transitions problem statement and its physical defines sense. Setting up them in a matrix receive the answer for this task.
5. Standard examples of creation of matrixes of transition give tasks about the wandering particles. Example. Let the system have five conditions of x1, x2, x3, x4, x5. The first and fifth are boundary. Let on each step the system can pass only into the state, next on number, and at the movement h5 about the probability of p, a h1 about the probability of q (p+q=1) aside aside. At achievement of borders the system can pass in h3 about the probability of v or remain in a former state with veroyatnostyyu1-v. Decision. In order that the task became absolutely transparent construct a state graph (see fig. 2).