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The probabilistic automaton may be defined as an extension of a nondeterministic finite automaton (,,,,), together with two probabilities: the probability of a particular state transition taking place, and with the initial state replaced by a stochastic vector giving the probability of the automaton being in a given initial state.
and Δt i = t i+1 − t i > 0, t 1 = 0 and t n = T. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes Δt i, but in the limit Δt i → 0 the probability density function becomes ill defined, one reason being that the product of terms
On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {X n}. The second term converges to zero as δ → 0, since the set B δ shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem.
The principle of maximum caliber (MaxCal) or maximum path entropy principle, suggested by E. T. Jaynes, [1] can be considered as a generalization of the principle of maximum entropy. It postulates that the most unbiased probability distribution of paths is the one that maximizes their Shannon entropy. This entropy of paths is sometimes called ...
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. [1]
1.6×10 −1: Gaussian distribution: probability of a value being more than 1 standard deviation from the mean on a specific side [20] 1.7×10 −1: Chance of rolling a '6' on a six-sided die: 4.2×10 −1: Probability of being dealt only one pair in poker 5.0×10 −1: Chance of getting a 'head' in a coin toss.
Seidel's algorithm is an algorithm designed by Raimund Seidel in 1992 for the all-pairs-shortest-path problem for undirected, unweighted, connected graphs. [1] It solves the problem in () expected time for a graph with vertices, where < is the exponent in the complexity () of matrix multiplication.
If X n converges in probability to X, and if P(| X n | ≤ b) = 1 for all n and some b, then X n converges in rth mean to X for all r ≥ 1. In other words, if X n converges in probability to X and all random variables X n are almost surely bounded above and below, then X n converges to X also in any rth mean. [10] Almost sure representation ...