Exponential Distribution

Definition

$X\sim \mathrm{Exp}(\lambda )={\Gamma }_{\alpha ,\beta }\left(1,\frac{1}{\lambda }\right)={\Gamma }_{k,\theta }(1,\lambda )$ where $\lambda \in {\mathbb{R}}^{+}$ is the average number of events per fixed unit of time

Time interval between events in a Poisson process with mean $\lambda$

Properties

PDF

$f(x)=\lambda e^{-\lambda x}\in {\mathbb{R}}_0^{+}$

CDF

$F(x)=1-e^{-\lambda x}$

Mean

$E(X)=\frac{1}{\lambda }$

Variance

$\mathrm{Var}(X)=\frac{1}{{\lambda }^2}$

Memorylessness Property

$\forall x,t\ge 0:P(T>s+t|T>s)=P(T>t)$