Memoryless

Memoryless is a term used in probability theory to describe a property where the future behavior of a system or random variable does not depend on past history beyond what is encoded in the present. For stochastic processes, memorylessness is often formalized by the Markov property: for any times s < t and any measurable set A, the probability that X_t lies in A, given the full history up to time s, depends only on the state at time s. In symbols, P(X_t ∈ A | F_s) = P(X_t ∈ A | X_s). Intuitively, once the present state is known, the past provides no additional information about the future. In continuous time this is frequently stated as the future being conditionally independent of the past given the present state.

Memoryless distributions are a related but special notion for individual random variables. A nonnegative random variable X is memoryless if for all s, t ≥ 0, P(X > s+t | X > s) = P(X > t). The only continuous distribution with this property is the exponential distribution; the only discrete nonnegative distribution with this property is the geometric distribution. Consequently, interarrival times in a Poisson process are exponentially distributed, and in discrete time, waiting times that are memoryless follow a geometric distribution.

Relation to Markov processes: the Markov property describes the dependence structure of future states given the present state, while memorylessness of a single variable is a stronger condition. A process can be Markov without every marginal distribution being memoryless. Nevertheless, memorylessness underpins many simple and tractable models, particularly in reliability, queueing theory, and stochastic processes.

Applications and examples include Poisson processes, M/M/1 queues, and reliability models where components have constant hazard rates. The concept helps explain why certain stochastic models yield tractable analysis and closed-form results.