A random process assigns a random variable to each time index.

For continuous time it can be written

$$X(t),$$

and for discrete time

$$X[n].$$

Before an outcome is observed, $X(t)$ is random at every time. After one experiment or realization, the process produces an ordinary signal called a sample path or realization.

For example, repeated measurements of thermal noise produce different voltage waveforms. Each waveform is one realization, while the random process describes the probability law governing the ensemble of possible waveforms.

Statistics can therefore depend on time. The mean function is

$$m_X(t)=\mathbb E[X(t)],$$

and pairs of times have joint statistics describing how values at different instants vary together.

Random processes provide the bridge between probability, which describes uncertain quantities, and signals, which evolve over time.