Exponential distribution

The exponential is the model of waiting times: how long passes before the next decay, the next arrival, the next failure. It is closely tied to the Poisson distribution (which counts the events) and to radioactive decay.

Property — $X\sim\text{Exp}(\lambda)$

Models the waiting times between events (decays, arrivals, failures). $f(x) = \lambda\,e^{-\lambda x} \text{ for } x\ge 0, \qquad f(x)=0 \text{ for } x<0.$ Mean $1/\lambda$, variance $1/\lambda^2$. The cumulative distribution function is $F(x) = 1-e^{-\lambda x}$.

Remark — Connection with ODEs

The exponential distribution is the same function that appears in the solution of the ODE $y'=-\lambda y$: radioactive decay. This is no coincidence: the survival probability $P(X>t) = e^{-\lambda t}$ satisfies exactly that differential equation.

Exponential distribution for three values of $\lambda$: large $\lambda$ means short waits (a curve that decays quickly, starting high), small $\lambda$ long waits.

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Topics: Distribuzioni probabilita
Concepts: Densita di probabilita · Distribuzione esponenziale · Funzione di ripartizione
Skills: Interpretare grafico · Modellizzare