probability and stochastic processes
... While many of the Galton’s viewpoints can be seen today as antiquated, or even offensive, the Galton–Watson branching process is still a central probabilistic tool applied in various fields of science. ...
... While many of the Galton’s viewpoints can be seen today as antiquated, or even offensive, the Galton–Watson branching process is still a central probabilistic tool applied in various fields of science. ...
A mini course on percolation theory
... 3. Uniqueness of the infinite cluster In terms of understanding the global picture of percolation, one of the most natural questions to ask, assuming that there is an infinite cluster, is how many infinite clusters are there? My understanding is that before this problem was solved, it was not comple ...
... 3. Uniqueness of the infinite cluster In terms of understanding the global picture of percolation, one of the most natural questions to ask, assuming that there is an infinite cluster, is how many infinite clusters are there? My understanding is that before this problem was solved, it was not comple ...
Probability Theory
... does . It is natural to believe that if → ∞ then # → 12 so that one says that ...
... does . It is natural to believe that if → ∞ then # → 12 so that one says that ...
MARKOV CHAINS: BASIC THEORY 1.1. Definition and First
... of the transition probability matrix on the simplex is a contraction? First, it tells us that if we start the Markov chain in two different initial distributions, then the distributions after one step are closer than they were to start. Consequently, by induction, after n steps they are even closer: ...
... of the transition probability matrix on the simplex is a contraction? First, it tells us that if we start the Markov chain in two different initial distributions, then the distributions after one step are closer than they were to start. Consequently, by induction, after n steps they are even closer: ...
- Wiley Online Library
... Recently, there has been a growing interest in another class of methods known as stochastic collocation method [12–14]. The basic idea of the stochastic collocation approach is to approximate the unknown stochastic solution by a polynomial interpolation function in the multi-dimensional random domai ...
... Recently, there has been a growing interest in another class of methods known as stochastic collocation method [12–14]. The basic idea of the stochastic collocation approach is to approximate the unknown stochastic solution by a polynomial interpolation function in the multi-dimensional random domai ...
More Efficient PAC-learning of DNF with Membership Queries Under
... Jackson [15] gave the first polynomial-time PAC learning algorithm for DNF with membership queries under the uniform distribution. However, the algorithm’s time and sample complexity make it impractical for all but relatively small problems. The algorithm is also not particularly efficient in its us ...
... Jackson [15] gave the first polynomial-time PAC learning algorithm for DNF with membership queries under the uniform distribution. However, the algorithm’s time and sample complexity make it impractical for all but relatively small problems. The algorithm is also not particularly efficient in its us ...
Randomness
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or ""trials"") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators.Random selection is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.