Stokastiska
Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noiseor the movement of a gas molecule. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, [ a ] used by Louis Bachelier to study price changes on the Paris Bourse[ 21 ] and the Poisson processused by A.
Erlang to study the number of phone calls occurring in a certain period of time. The term random function is also used to refer to a stochastic or random process, [ 25 ] [ 26 ] because a stochastic process can also be interpreted as a random element in a function space. Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks[ 31 ] martingales[ 32 ] Markov processes[ 33 ] Lévy processes[ 34 ] Gaussian processes[ 35 ] random fields, [ 36 ] renewal processesand branching processes.
A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. Historically, the index set was some subset of the real stokastiskasuch as the natural numbersgiving the index set the interpretation of time.
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time.
The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. If the stokastiska space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space.
The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year as its earliest occurrence. The term stochastic process first appeared in English in a paper by Joseph Doob. According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence in riding, running, striking, etc.
The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop".
Kort om sannolikhetsfunktioner och fördelningsfunktioner för
The first written appearance of the term random process pre-dates stochastic processwhich stokastiska Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in The definition of a stochastic process varies, [ 67 ] but a stochastic process is traditionally defined as a collection of random variables indexed by some set. The term random function is also used to refer to a stochastic or random process, [ 5 ] stokastiska 74 ] [ 75 ] though sometimes it is only used when the stochastic process takes real values.
Random walks are stochastic processes that are usually defined as sums stokastiska iid random variables or random vectors in Euclidean stokastiska, so they are processes that change in discrete time. A classic example of a random walk is known as the simple random walkwhich is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one.
Stochastic - Wikipedia
The Wiener process is a stochastic process with stokastiska and independent increments that are normally distributed based on the size of the increments. Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.
Almost surelya sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. The Poisson process is a stochastic process that has different forms and definitions.
The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. Stokastiska on the real line, the Poisson process can be interpreted as a stochastic process, [ 49 ] [ ] among other random objects.
There are other ways to consider a stochastic process, with the above definition being considered the traditional one. The state space is defined using elements that reflect the different values that the stochastic process can take.
Stokastiska variabler & listbarhet
A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index stokastiska that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period.
Stokastiska law of a stochastic process or a random variable is also called the probability lawprobability distributionor the distribution.