Hey ! .. what is a stochastic process ?
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In probability theory and related fields, a stochastic 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.
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A stochastic process is simply a random process through time. A good way to think about it, is that a stochastic process is the opposite of a deterministic process.
In a deterministic process, given the initial conditions and the parameters of the system, we can define the exact "position" of the system at any time. In a stochastic process, we don't know where the process will be, even if we know the initial conditions and parameters.
In a formally defined stochastic process, such as a random walk, or a game of chance (like blackjack for instance), we can infer certain characteristics of the process, such as the expected value and variance. These are formally defined because the parameters of the process are known. We can write down the structure of the process on a piece of paper. For example, take a simple random walk. The process starts at 0. There's a 50:50 chance the process goes up or down at each time step. The expected value of the process is always zero, however the variance of the process expands with each step. After 100 steps we expect to be at 0 but we could easily be at 100 (although unlikely).
Stochastic models are used to attempt to model complex and dynamical systems, such as the stock exchange or weather systems. These systems are not inherently random, but they're sufficiently complicated, and incomprehensible to us, that we could never model them as deterministic, so we use stochastic models to summarise them.
A real life example? How about is a pollen grain floating in water. There's some wonderfully history behind that. I encourage you to check it out!
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