Concept

Exponential Distribution

The exponential distribution models waiting time to the next event in a Poisson process.

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exponential distribution

Formula

If T is exponential with rate lambda, the survival probability decays exponentially with time.

Survival function

P(T>t)=e^{-\lambda t}

Mean and variance

E[T]=\frac{1}{\lambda},\qquad \operatorname{Var}(T)=\frac{1}{\lambda^2}

If claim arrivals average 0.5 per day, the chance the next claim takes more than 3 days is e^(-0.5 x 3), about 0.223.

The memoryless property is about remaining waiting time, not about the event becoming impossible or guaranteed after time passes.