Review Stochastic Process

A stochastic process is a collection of random variables indexed by time, space, or another set. It models how an uncertain quantity or system can vary across that index under a probability law. It is also commonly called a random process.

Think of a sequence of snapshots taken from something that can change unpredictably. Each snapshot has possible values, and the process describes both those possibilities and how one snapshot may relate to another. One observed sequence is a single realization of the process, not the entire process.

A stochastic-process view helps people reason about systems whose behavior varies from one observation or run to the next. Instead of treating one outcome as the rule, they can examine distributions, dependencies, trends, and ranges across many observations. This matters when evaluating queues, demand, sensor readings, financial values, and some AI systems that can produce different outputs across repeated runs.

Suppose a team records the number of requests waiting in a service queue at the end of every minute. Let each recorded queue length be a random variable, because arrivals and completion times vary. The indexed collection of those random variables is a stochastic process. One day's sequence of observed queue lengths is one realization of that process.

“Stochastic” does not mean completely unpredictable, and it does not mean that every observation is independent. A stochastic process can have strong patterns, limits, trends, or dependencies while still including uncertainty.

A random variable is one uncertain snapshot. A stochastic process is an indexed series of uncertain snapshots together with the probability structure connecting them.

* [[random-process|Random Process]] * [[process|Process]] * [[observation|Observation]] * [[system|System]]

* [MIT OpenCourseWare, “Lecture 5: Stochastic Processes I”](https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/f5784e4facf3de690210d17c97358eba_MIT18_S096F13_lecnote5.pdf) — definition as indexed random variables and the sample-path view. * [Princeton University, “Statistical Signal Processing: Random Process”](https://www.princeton.edu/~cuff/ele530/lectures/processes.html) — random-process terminology, index sets, and temporal or spatial examples. * [NIST/SEMATECH e-Handbook, “The process is a statistical process”](https://www.itl.nist.gov/div898/handbook/pmd/section2/pmd211.htm) — the role of random variation in statistical process modeling.

* [MIT OpenCourseWare: Topics in Mathematics with Applications in Finance](https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/) * [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)