exponential distribution

An exponential random variable is a continuous random variable modeling the distance between occurrences in a Poisson distribution. Such a random variable follows an exponential distribution, denoted $X\sim \mathrm{Exp}(\lambda )$, where $\lambda >0$ is the rate parameter.

Formally,
if $X$ is an exponential random variable, i.e. $X\sim \mathrm{Exp}(\lambda )$, then its probability density function is

and its cumulative distribution function is

both defined for $x\ge 0$, with probability zero otherwise.

If $X\sim \mathrm{Exp}(\lambda )$, then

• the expected value $E[X]=\frac{1}{\lambda }$
• the variance $\mathrm{Var}(X)=\frac{1}{{\lambda }^2}$
• the standard deviation $\sigma =\frac{1}{\lambda }$