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Why do we study stochastic process?
Stochastic processes underlie many ideas in statistics such as time series, markov chains, markov processes, bayesian estimation algorithms (e.g., Metropolis-Hastings) etc. Thus, a study of stochastic processes will be useful in two ways: Enable you to develop models for situations of interest to you.
What is stochastic function?
A stochastic (random) function X(t) is a many-valued numerical function of an independent argument t, whose value for any fixed value t ∈ T (where T is the domain of the argument) is a random variable, called a cut set . In this case, the function X(t) is called a stochastic (random) field .
Which is the covariance function of a stochastic process?
Usually, for a stochastic process X, CX is called the autocovariance function of the process and the term covariance is reserved for the correlation between two different stochastic processes.
How to characterize a stochastic process in PDF?
How to characterize a stochastic process: Use n-dimensional pdf (or cdf or pmf) of n random variable at n randomly selected time instants. (It is also called nth-order pdf). Generally, the n-dimensional pdf is time varying. If it is time invariant, the stochastic process is stationary in the strict sense.
When is X said to be stationary by stochastic processes?
More generally, X is said to be stationary if p(X(t 1), ⋯, X(t n)) = p(X(t 1 + τ), ⋯, X(t n + τ)) for all, τ, i.e., the joint probabilities of X at different times are independent of the reference point τ. This implies in particular that both the mean and autocovariance functions are independent of the reference time point.
What are the joint distributions of a stochastic process?
In practice, a random process is characterized by the set of its joint distributions p ( X (μ, t1 ), …, X (μ, tn )) of values taken at fixed times t1, tn. Stochastic processes are thus a direct generalization of random vectors as defined in §11.9.