Exponentialdiskontierung

The exponential distribution is a continuous probability distribution that describes the waiting time between events in a Poisson process with a constant average rate. It is defined for nonnegative real numbers.

Its probability density function is f(x) = λ e^(−λx) for x ≥ 0, where λ > 0 is the rate parameter. The cumulative distribution function is F(x) = 1 − e^(−λx). Equivalently, with the scale parameter θ = 1/λ, the density can be written as f(x) = (1/θ) e^(−x/θ) for x ≥ 0. The mean is 1/λ (or θ) and the variance is 1/λ^2 (or θ^2).

A key feature of the exponential distribution is its memoryless property: for all s, t ≥ 0, P(X > s + t | X > s) = P(X > t). Its moment generating function is M(t) = λ/(λ − t) for t < λ, confirming its relation to the gamma family with shape k = 1.

Relation to other distributions: the exponential distribution is a special case of the gamma distribution with shape parameter 1. It is often regarded as the continuous analogue of the geometric distribution on waiting times.

Parameter estimation and usage: in data, the maximum likelihood estimate of λ is λ̂ = n / ∑x_i, equivalently 1/ x̄ for a sample {x_i}. The exponential distribution is widely used to model lifetimes, service times, and interarrival times in reliability engineering, queuing theory, and risk assessment due to its simplicity and memoryless property.