The exponential distribution with rate ${\displaystyle \lambda }$, written ${\displaystyle \operatorname {Exp} (\lambda )}$,^[1] is a probability distribution defined only on the positive real numbers. It is the continuous analogue of geometric distribution.^[2] Exponential distribution is used in reliability applications,^[3] and its main use is to assess the duration of random time intervals. Examples where it can be used include:

• Time length of a telephone call
• How long does it take to perform a service (for example, to fix something at a service point)
• Amount of time between two phone calls
• Half life of atoms (radioactive decay)
• Expected lifetime of electronic (or other) parts, if wearing is not considered (this is called Mean Time Between Failures, MTBF)
• Age of plants or animals
• Very simple model used by insurance companies