Stochastic Process¶

Definition¶

Let $T$ be an index set. A stochastic process is a collection of random variables $\{X_t\colon t\in T\}$ defined on a common probability space, that is, one sample space with one probability function.

Interpretation¶

A stochastic process describes randomness that develops over time, or over another index. Instead of one random value, it is a whole family of random values $X_t$, one for each index $t$, all driven by one random experiment.

The two zigzag lines are sample paths for two outcomes $\omega_1$ and $\omega_2$; at each step the value moves up or down by $1$. Fixing the outcome $\omega$ gives a function of time, $t\mapsto X_t(\omega)$$X_t(\omega )$. The vertical line at $t=6$ does the opposite: fixing the time gives the random variable $X_6$, which takes the value $2$ on the blue path and $-2$ on the red one.

Defining all $X_t$ on one probability space is what makes statements about several times at once meaningful, such as $\mathbb{P}(X_5>0\text{ and }X_6>0)$ or the probability that a path reaches level $4$ before time $10$. The distributions of the single $X_t$ do not determine such probabilities; the dependence between different times matters as well.

With $T=\{0,1,2,\ldots\}$ the process runs in discrete time and each path is a sequence, as in the figure. With $T=[0,\infty)$ it runs in continuous time, like the count $N_t$ of a Poisson process, whose paths are staircases that rise by one at each arrival.

A process is not a single random variable that changes over time: each $X_t$ is a separate random variable, and a path is one realization of the whole family, not the process itself.

Key Results¶

• A stochastic process is a family of random variables indexed by time or another set.
• The index set may be discrete or continuous.
• Sample paths describe how one realization of the process evolves across the index set.

Examples¶

• Daily demand over a year can be modelled as a discrete-time stochastic process.
• The number of arrivals up to time $t$ in a queue is a continuous-time stochastic process.
• Stock prices observed over time form a stochastic process.

Prerequisites¶

• Random Variable

Related Concepts¶

• Arrival Rate
• Matrix
• Sequence
• Function

Sources¶

• Probability Theory for EOR · Week 6

Aliases: random process Keywords: collection of random variables Symbols: element of, infinity, omega

Last updated: 20 October 2026 03:02