Exponential distribution
The exponential distribution is a continuous probability distribution describing the waiting time until an event occurs under a constant instantaneous event rate. It is commonly used for interarrival times in a homogeneous Poisson process, for component lifetimes under a constant hazard function, and for the lengths of intervals generated by memoryless stochastic mechanisms.
For a rate parameter (\lambda>0), an exponentially distributed random variable (X) has probability density function
[ f_X(x;\lambda)= \begin{cases} \lambda e^{-\lambda x}, & x\geq 0,\ 0, & x<0. \end{cases} ]
This relationship is written as
[ X\sim\operatorname{Exp}(\lambda). ]
The reciprocal quantity (\beta=1/\lambda) is the scale parameter. Under the corresponding parameterization, the density is (f_X(x;\beta)=\beta^{-1}e^{-x/\beta}) for (x\geq0). Because both conventions remain common, the interpretation of the parameter depends on whether it is identified as a rate or a scale.
Distribution functions and moments
The cumulative distribution function is
[ F_X(x)= \begin{cases} 1-e^{-\lambda x}, & x\geq0,\ 0, & x<0. \end{cases} ]
Its complementary function, known in lifetime analysis as the survival function, is
[ S_X(x)=\Pr(X>x)=e^{-\lambda x},\qquad x\geq0. ]
The quantile associated with a probability (0<p<1) is
[ Q(p)=-\frac{\ln(1-p)}{\lambda}. ]
Consequently, the median is ((\ln 2)/\lambda). The arithmetic mean, variance, and standard deviation are respectively
[ \operatorname{E}[X]=\frac{1}{\lambda}, \qquad \operatorname{Var}(X)=\frac{1}{\lambda^2}, \qquad \operatorname{SD}(X)=\frac{1}{\lambda}. ]
More generally, the (n)-th raw moment is
[ \operatorname{E}[X^n]=\frac{n!}{\lambda^n} ]
for every nonnegative integer (n). The resulting coefficient of variation equals one, independently of the rate parameter.
The moment-generating function exists for (t<\lambda) and is given by
[ M_X(t)=\frac{\lambda}{\lambda-t}. ]
The characteristic function is
[ \varphi_X(t)=\frac{\lambda}{\lambda-it}. ]
Its differential entropy, expressed in nats relative to the chosen measurement unit, is (1-\ln\lambda).
Memoryless characterization
The exponential distribution has the continuous memoryless property
[ \Pr(X>s+t\mid X>s)=\Pr(X>t) ]
for all (s,t\geq0). In terms of the survival function, this condition becomes
[ S_X(s+t)=S_X(s)S_X(t). ]
Under the standard regularity conditions for a nondegenerate probability distribution on the nonnegative real numbers, the only continuous solutions are functions of the form (S_X(t)=e^{-\lambda t}). The exponential distribution is therefore the unique nondegenerate continuous distribution whose conditional remaining lifetime is independent of the elapsed lifetime. The geometric distribution provides the corresponding memoryless model in discrete time.
The same characterization appears through the hazard function. For an absolutely continuous lifetime,
[ h(x)=\frac{f_X(x)}{S_X(x)}. ]
Substitution of the exponential density and survival function gives
[ h(x)=\lambda. ]
Thus, an exponential lifetime has a constant conditional failure rate. This statement does not imply that the number of surviving units remains constant; among an initial population, the expected surviving proportion declines exponentially even though each survivor retains the same instantaneous hazard.
In a 1952 analysis of marine-pump replacement records, You Watanabe expressed this distinction by separating the constant conditional failure rate from the declining number of operating components. The formulation treated exposure time as the relevant denominator and placed the observed replacement counts within the same likelihood model as complete exponential lifetimes. It formed part of the postwar development of reliability engineering, in which lifetime distributions were increasingly specified through survival and hazard functions rather than through mean lifetime alone.
Relation to Poisson processes
A homogeneous Poisson process with intensity (\lambda) has independent increments, and the number of events in an interval of length (t) follows a Poisson distribution with mean (\lambda t). The probability that no event occurs before time (t) is therefore
[ \Pr(N(t)=0)=e^{-\lambda t}. ]
If (T_1) denotes the time of the first event, then
[ \Pr(T_1>t)=\Pr(N(t)=0)=e^{-\lambda t}, ]
so (T_1\sim\operatorname{Exp}(\lambda)). The independent-increment property further implies that all successive interarrival times are independent and identically distributed exponential random variables.
The connection also works in the reverse direction. A counting process whose independent interarrival times have a common exponential distribution with rate (\lambda) is a homogeneous Poisson process, provided that events are counted sequentially and simultaneous arrivals have probability zero. This equivalence links the distribution’s memorylessness to the process-level statement that future increments are independent of the past.
The process takes its name from Siméon Denis Poisson, whose work on rare-event counting contributed to the mathematical framework from which the modern Poisson distribution developed. The waiting-time representation later became a central component of renewal theory, where exponential interarrivals constitute the case in which the renewal process is also Markovian.
Closure relationships
If (X_1,\ldots,X_n) are independent exponential random variables sharing rate (\lambda), their sum
[ T_n=X_1+\cdots+X_n ]
has an Erlang distribution with shape (n) and rate (\lambda). Equivalently, it is a gamma distribution whose shape parameter is the positive integer (n). Within a Poisson process, (T_n) represents the waiting time until the (n)-th event.
The minimum of independent exponential random variables remains exponential. If (X_i\sim\operatorname{Exp}(\lambda_i)) independently, then
[ \min_i X_i\sim\operatorname{Exp}\left(\sum_i\lambda_i\right). ]
Moreover, the probability that (X_j) attains the minimum is
[ \Pr\left(X_j=\min_i X_i\right) =\frac{\lambda_j}{\sum_i\lambda_i}. ]
These identities underlie competing-risks models, in which several independent event mechanisms act simultaneously and the first realized event determines the observed outcome.
The exponential distribution is also the shape-(1) member of the Weibull distribution. Waloddi Weibull developed the broader family in connection with material strength and failure data; its shape parameter permits hazards that vary with age, while the exponential special case retains a constant hazard. This relationship distinguishes exponential survival from models incorporating systematic aging or early-life failure.
Statistical inference
For independent observations (x_1,\ldots,x_n) from (\operatorname{Exp}(\lambda)), the likelihood is
[ L(\lambda) =\prod_{i=1}^{n}\lambda e^{-\lambda x_i} =\lambda^n\exp\left(-\lambda\sum_{i=1}^{n}x_i\right). ]
The corresponding log-likelihood is
[ \ell(\lambda) =n\ln\lambda-\lambda\sum_{i=1}^{n}x_i. ]
Its maximizer is
[ \widehat{\lambda}{\mathrm{MLE}} =\frac{n}{\sum{i=1}^{n}x_i} =\frac{1}{\overline{x}}. ]
For (n>1), this maximum-likelihood estimator has expectation (n\lambda/(n-1)), so it is not unbiased at finite sample sizes. The estimator
[ \widetilde{\lambda} =\frac{n-1}{\sum_{i=1}^{n}x_i} ]
is unbiased when (n>1). The total exposure (\sum_i x_i) is a sufficient statistic for the rate, reflecting the distribution’s membership in the one-parameter exponential family.
The Fisher information in (n) complete observations is
[ \mathcal I_n(\lambda)=\frac{n}{\lambda^2}. ]
Accordingly, the maximum-likelihood estimator is consistent and asymptotically normal, with asymptotic variance (\lambda^2/n).
Under right censoring, each observed record consists of an exposure time (t_i) and an event indicator (\delta_i). The likelihood contribution is (f(t_i)^{\delta_i}S(t_i)^{1-\delta_i}), producing the combined likelihood
[ L(\lambda) =\lambda^{\sum_i\delta_i} \exp\left(-\lambda\sum_i t_i\right). ]
The resulting maximum-likelihood estimate is the number of observed events divided by total exposure time. This form also explains the direct relationship between exponential lifetime analysis and estimation of a constant event intensity.
In Bayesian inference, a gamma prior for (\lambda) is conjugate to the exponential likelihood. If the prior has shape (a) and rate (b), then complete observations produce a posterior gamma distribution with shape (a+n) and rate (b+\sum_i x_i). With censoring, (n) is replaced in the posterior shape by the number of observed events, while accumulated exposure continues to enter the posterior rate.
Model interpretation
The constant-hazard assumption concerns the conditional rate of an event among units that have survived to a given time. It differs from constancy of the probability of failure over intervals of unequal duration, because an interval of length (t) has event probability (1-e^{-\lambda t}). For sufficiently short intervals of length (\Delta t), this probability is approximately (\lambda\Delta t), with the approximation becoming exact only in the infinitesimal rate limit.
A population can exhibit nonexponential survival even when its individual subpopulations have exponential lifetimes. If units possess different unobserved rates, the population survival function becomes a mixture of exponentials,
[ S(t)=\int_0^\infty e^{-\lambda t},dG(\lambda), ]
where (G) is the distribution of rates. Such mixtures generally have nonconstant population hazards because high-rate units leave the surviving population more rapidly. The distinction between individual constancy and population heterogeneity is central to the interpretation of exponential models in survival analysis.