exponential distribution
- Related Topics:
- Poisson distribution
- distribution function
exponential distribution, a continuous probability distribution used to determine the time taken by a continuous process, occurring at an average rate, to change its state.
The exponential distribution is used to model situations in which smaller values are more likely to occur than larger values. Scenarios such as durations of phone calls and amounts of change in people’s pockets can be modeled using the exponential distribution, since lower values occur more often, with a decline in probability as values increase.
The exponential distribution is memoryless: the time of the next event does not depend on how much time has elapsed since the previous event. Each event in an exponential distribution is thus independent of other events. The memoryless nature assumes that the average time between events remains constant. The exponential distribution therefore cannot be used in situations in which the time between events changes, such as when predicting machine failures, as machines wear out over time, causing the failure rate to not stay constant but to increase as time passes.
Mathematical formula
The formula to calculate the probability density function (PDF), or simply probability, of an exponential distribution isf(x) = λe−λx for x > 0,
where λ is the distribution rate, or the number of events per unit time.
The average value of the exponential distribution isE(x) = 1/λ.
The variance of the exponential distribution isVar(x) = 1/λ2.
Similar distributions
The exponential distribution can be seen as a continuous version of the geometric distribution, which is also memoryless. A geometric distribution looks at events with two outcomes and models the probability of achieving success across independent trials. The exponential distribution can, in this scenario, help determine the time at which the first success will be achieved.
The exponential distribution is also closely related to the Poisson distribution, in that the Poisson distribution helps determine when a random event will occur, given that the average rate of events is constant. The exponential distribution will then model the time between these events. The gamma distribution is similar, but, although the exponential distribution looks at modeling time between events, the gamma distribution models the time until a certain number of events occurs.
Real-life applications
Exponential distributions help in reliability testing to determine failure rates (but only in cases in which the rate of failure stays constant and independent of past failure events). They are used in science to calculate decay times of radioactive particles and predict geyser eruptions, as well as approximate earthquake arrival times.