Stochastic processes
A stochastic process is a collection of random variables indexed by time, commonly written
\[\{X(t):t\in T\}.\]For each time \(t\), \(X(t)\) is a random variable. One realised sequence through time is called a sample path or trajectory.
State and time
Time may be discrete or continuous, and the state space may also be discrete or continuous. These choices produce different classes of stochastic models.
Biological example
If \(I(t)\) is the number of infectious individuals at time \(t\), repeated epidemics can generate different paths even when they start from the same \(I(0)\).
Key idea. A stochastic process describes how a random state evolves through time, not merely one random observation.