Exponential distribution

A continuous probability distribution for nonnegative waiting times between events in a Poisson process with a constant rate. Its defining memoryless property means the expected remaining wait does not depend on time already elapsed.

Probability density function: A function whose integral over an interval gives the probability that a continuous random variable lies there. The exponential density assigns probabilities to ranges of waiting times.
Poisson process: A counting process with independent, stationary increments and a constant average event rate. The exponential distribution describes intervals between successive events in this process.
Reliability engineering: The engineering discipline concerned with the ability of systems to perform without failure over time. Exponential lifetimes model components whose failure rate remains constant.
Weibull distribution: A continuous distribution often used for lifetimes, with a hazard that can increase, decrease, or remain constant. Unlike the exponential model, it can represent changing failure rates over time.
Hazard function: A function describing the instantaneous event rate conditional on survival up to a given time. Its constant value is the key rate property of exponential waiting times.
Poisson distribution: A discrete probability distribution for counts of events in a fixed interval under a constant-rate process. Poisson counts and exponential waiting times are two descriptions of the same event process.
Queueing theory: The mathematical study of waiting lines, service systems, and their performance. Exponential interarrival and service times make many queueing models analytically tractable.
Gamma distribution: A continuous distribution on positive values that generalizes the exponential distribution through a shape parameter. It models waiting until multiple Poisson events rather than just the next event.
Survival function: The probability that a random variable exceeds a specified value. For exponential waiting times, survival declines exponentially with elapsed time.
Conditional probability: The probability of an event given that another event has occurred. The memoryless property is expressed as a particular identity involving conditional probabilities.

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