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I'm a software engineering student, so I don't have probability theory in my career, but I hear people near my circle talk about stochastic processes very often. Could someone explain in simple words what these are?

E: I'll make myself a bit more precise. I've heard people describe a series of events via a stochastic process, I've read the wiki, and I don't understand how the 'collection of random variables' part comes to play: what does it mean to represent something via a non-deterministic process?

Could somebody provide a concrete example of a representation of something via a stochastic process (and how it defers from a deterministic process)?

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  • $\begingroup$ math.stackexchange.com/q/885349 $\endgroup$
    – Did
    Oct 8, 2015 at 17:24
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    $\begingroup$ A stochastic process is a colection of random variables defined on the same probability space. Please explain further what parts of this definition are escaping you. $\endgroup$
    – Did
    Oct 8, 2015 at 17:25
  • $\begingroup$ @Did Done so, thanks for the suggestion. $\endgroup$ Oct 8, 2015 at 17:30

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Very roughly speaking, you can think of a stochastic process as a process that evolves in a random way. The randomness can be involved in when the process evolves, and also how it evolves.

A very simple example of a stochastic process is the decay of a radioactive sample (with only one parent and one daughter product). Initially, it has some large number $N$ of atoms of the parent element. Over time, the number of such atoms decreases, always by $1$, but at random moments in time. The state of the system can be represented by $k$, the number of atoms of the parent element present at a given moment in time. Initially, $k = N$, but eventually, it will fall to zero.

In this process, when the state changes is random, but not how it changes. In other processes, such as a discrete-time random walk, when the state changes is deterministic, but how it changes is random. And there are other processes in which both when the state changes and how it changes are random.

Interestingly, in many cases, stochastic processes are used to model situations that may not have inherent randomness. For instance, Brownian motion is the result of forces that could, in principle, be determined precisely (if we ignored quantum mechanics). However, the number of objects in a normal system is so large that such an analysis would be intractable. Instead, we model the motion of objects using a stochastic process, and thereby obtain some insight into the behavior of such systems (for instance, the statistical behavior of a given particle over time) that we could not begin to with a deterministic approach.

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    $\begingroup$ I like the point about stochastic processes being used to model deterministic situations that are simply too complicated to fully understand. Even when rolling dice and flipping coins, we could figure out the answer if we knew all the physics involved, but it's simpler to treat the outcomes as random. $\endgroup$
    – eigenchris
    Oct 8, 2015 at 20:18
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A stochastic process is a way of representing the evolution of some situation that can be characterized mathematically (by numbers, points in a graph, etc.) over time.

They are of greatest help when you either don't know the exact rules of that evolution over time, or when the exact rule of that evolution is too complicated or costly to compute precisely.

Instead of trying to compute the exact evolution of the system, you use a source of randomness to help you describe the situation and its evolution. Then, using the laws of probability, you may be able to compute an expected behavior over time, the probability that something desirable happens, whether the situation leads to some stable state, etc.

A typical concrete example is the length of a queue waiting for a cashier over time. Knowledge of the exact evolution of this number over the day could come in handy to the administrators, but they don't have exact knowledge of

  1. What makes people come at the exact times that they do, or
  2. How many items they will bring, or
  3. Any exceptional situations which could slow down the cashier.

If the number of people in the queue at time $t$ is $N_t$, then we could consider each $N_t$ to be a random variable, because we don't know for sure what will happen at that moment.

Randomness does not necessarily imply chaotic or "crazy" behavior, it can also obey its own laws. For example, if $N_t=5$, then we expect it to stay at $5$ in the moments following $t$ until someone arrives or leaves the queue at some time $t+s$, and then it can only jump to the values $N_{t+s} = 6$ or $4$.

In this case, something similar to a birth and death process could represent the situation. In this process there is randomness only in the amount of time that passes between changes in the state of $N$ and in the direction in which the state changes ($\pm 1$); not in its magnitude.

Of course, in order for a stochastic process to accurately represent a given situation, its underlying assumptions must me compatible with the situation, even if only as an approximation. The modeling process may involve estimating parameters, testing hypotheses, etc.

Famous examples of stochastic processes are Brownian motion, random walk, the Black Scholes model for financial derivatives and the Poisson process.

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I'd like to add that many types of noise are also modeled as stochastic processes. Whenever a person collects data, such a voltage signals, audio signals, or image signals, there will always be small disturbances in the data caused by imperfections in the equipment or the environment. The signal we measure might be given by the formula:

$$measured\ signal = true\ signal + noise $$

One famous example you may have heard of is white noise, which is a stochastic process where the signal value at any given time is completely decorrelated with past and future signal values.

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It is inherently random or noisy - not deterministic. Check out the wikipedia page.

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    $\begingroup$ True--but hardly addressing the question. $\endgroup$
    – Did
    Oct 8, 2015 at 17:26
  • $\begingroup$ I would say that this exactly addresses the question. As a software engineer, this means that the algorithm is nondeterministic (assuming the random number generators are not seeded with the same value), and you cannot rely on the answers being the same from run to run. $\endgroup$ Oct 8, 2015 at 17:28
  • $\begingroup$ @NoseKnowsAll Thus, you and this answer are (somewhat) addressing the question What is a random variable? This is interesting on its own and might even be what the OP actually wanted to ask, but this is not what they are asking. $\endgroup$
    – Did
    Oct 8, 2015 at 17:32

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