mathematics//statistics//exponential distribution

The exponential distribution is the probability distribution of the waiting time until an event that has the same chance of happening in every instant, whatever has happened before, and it is the model of lifetimes with a constant failure rate (electronics hit by random surges, parts broken by external shocks) and of the gaps between arrivals that occur at random. With rate \(\lambda\) events per hour, the probability of surviving beyond \(t\) and the mean waiting time are

The exponential distribution is the probability distribution of the waiting time until an event that has the same chance of happening in every instant, whatever has happened before, and it is the model of lifetimes with a constant failure rate (electronics hit by random surges, parts broken by external shocks) and of the gaps between arrivals that occur at random. With rate λ\lambdaλ events per hour, the probability of surviving beyond ttt and the mean waiting time are

R(t)=P(T>t)=e−λt,E[T]=1λ.R(t)=P(T>t)=e^{-\lambda t},\qquad \mathbb E[T]=\frac1\lambda .R(t)=P(T>t)=e−λt,E[T]=λ1​.

A board with λ=10−5\lambda=10^{-5}λ=10−5 failures per hour has a mean time between failures (MTBF) of 100,000 h, and only e−1≈37%e^{-1}\approx37%e−1≈37% of a population survives to that mean: most units die before the average life.

Its defining property is memorylessness. The chance of failing in the next hour is the same for a unit with 10,000 hours of service as for one installed this morning, because the hazard function (the failure rate of survivors) is the constant λ\lambdaλ. It is the only continuous distribution with that property, and in the language of Weibull distribution it is the case of shape β=1\beta=1β=1, the flat bottom of the bathtub curve.

With a constant hazard, preventive replacement only spends spares.

The part taken out is exactly as likely to fail tomorrow as the new one put in, and the new one may add infant-mortality failures of its own. Replacing on a calendar pays only when the hazard rises with age (wear, fatigue, corrosion); for random failures the tools are redundancy, spares on the shelf and fast repair (preventive maintenance).

Its rate is easy to estimate, and the estimate needs failures. The maximum likelihood estimate of λ\lambdaλ is the number of failures over the total unit-hours observed, survivors included; with zero failures only an upper bound exists (sample size, rule of three).

Random arrivals have exponential gaps. Requests reaching a server, robots arriving at a charger or calls on a line, when each comes independently of the others, form a Poisson process whose inter-arrival times are exponential; it is the M in the M/M/1 queue (point process).

It is often assumed because it is convenient. Reliability handbooks that quote one failure rate per component assume it; field data for mechanical parts rarely agree, and a fit of the Weibull distribution says which regime a population is in.