Contents
What is meant by stochastic variables?
Stochastic Variables First the concept of the stochastic (or random) variable: it is a variable X which can have a value in a certain set Ω, usually called “range,” “set of states,” “sample space,” or “phase space,” with a certain probability distribution.
What is stochastic series?
In probability theory and related fields, a stochastic (/stoʊˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.
Is every stochastic process a time series?
The stochastic process is considered to generate the infinite collection (called the ensemble) of all possible time series that might have been observed. Every member of the ensemble is a possible realization of the stochastic process. A set of observed time series is considered to be a sample of the population.
Are time series random variables?
A time series is a collection of random variables indexed by time, for example X(1), …, X(n). In particular, the random variables can be dependent and their distribution might change over time.
What’s the difference between time series and stochastic process?
A time series is a stochastic process that operates in continuous state space and discrete time set. A stochastic process is nothing but a set of random variables. It is a time dependent random phenomenon.
How is a stochastic process represented in probability theory?
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers.
Is the random variable in a stochastic process always a number?
If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.
When does a stochastic process have continuous state space?
Classifications. If the state 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.