The exponential distribution is a probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.
The probability density function (PDF) for the exponential distribution is given by f(x) = λe^(-λx), where λ is the rate parameter and x is the random variable.
The exponential distribution is characterized by its mean (1/λ), variance (1/λ^2), and standard deviation (1/λ). It is a continuous distribution with a range of [0, ∞), and its shape is skewed to the right.
The exponential distribution is commonly used in reliability and survival analysis to model the time until failure of a component or the time until death of an organism. It is also used in queueing theory to model waiting times in systems with continuous arrivals.
The exponential distribution is closely related to the Poisson distribution, as the time between events in a Poisson process follows an exponential distribution. This means that if the number of events in a given time period follows a Poisson distribution, the time between those events will follow an exponential distribution.