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Exponential(λ) with rate λ>0: f(x)=λe−λx, x≥0. E[X]=1/λ, Var(X)=1/λ². Models lifetimes, service times, and waiting times between Poisson events. Only continuous distribution with memoryless property.
From your introduction to the exponential distribution, you know the basic shape and the formulas. This deeper treatment develops two things: why the exponential distribution is the unique continuous distribution with the memoryless property, and what it means for the exponential to be the continuous-time analogue of the geometric distribution — both arising as waiting times in a Poisson framework.
The memoryless property says: P(X > s + t | X > s) = P(X > t) for all s, t ≥ 0. In words: if a component has survived to age s, its remaining lifetime has exactly the same distribution as a brand-new component. Past survival gives no information about future failure. This is a strange property — it means the exponential distribution has no aging. Formally, the only continuous distribution with this property is the exponential. Here is the argument: the survival function S(x) = P(X > x) must satisfy S(s + t) = S(s) · S(t) (this is the functional equation the memoryless property imposes). Continuous solutions to S(s + t) = S(s) · S(t) with S(0) = 1 and S decreasing are exactly S(x) = e−λx for some λ > 0 — i.e., the exponential distributions. No other continuous distribution satisfies this.
The connection to the Poisson process is where the exponential becomes indispensable. If events occur as a Poisson process with rate λ (meaning the number of events in any interval of length t is Poisson(λt)), then the waiting time between consecutive events is Exp(λ). Conversely, if inter-arrival times are i.i.d. Exp(λ), the counting process is Poisson with rate λ. This duality means the exponential and Poisson distributions are two views of the same underlying random process: Poisson describes the count, exponential describes the gaps. The mean waiting time 1/λ is the reciprocal of the rate, which is intuitive: if events arrive at rate 2 per hour (λ = 2), the average wait is 1/2 hour.
For practical calculations: the CDF is F(x) = 1 − e−λx, making probability computations straightforward. Sums of independent exponentials produce the gamma distribution: if X₁, …, Xₙ are i.i.d. Exp(λ), then X₁ + … + Xₙ ~ Gamma(n, λ). The minimum of independent exponentials is again exponential: min(X₁, X₂) ~ Exp(λ₁ + λ₂) when Xᵢ ~ Exp(λᵢ) independently — a crucial fact in reliability theory and queueing, where the system fails when the first component fails. Together, these properties — memorylessness, Poisson duality, additive gamma structure, and minimum closure — make the exponential the cornerstone of continuous-time probability models.