xbeta
Exponential distribution

Characterization

The probability density function? (pdf) of an exponential distribution with rate parameter $\lambda$ is

The cumulative distribution function? (cdf) is

The distribution has support $[0,\mathrm{\infty })$. If a random variable $X$ has this distribution, we write $X\sim Expo(\lambda ).$

Alternative parameterization

Sometimes the exponential distribution is parameterized by $\beta =\frac{1}{\lambda }$ where $\beta$ is the mean of the $Expo(\lambda )$ distribution. Thus, to avoid ambiguity it is important to specify whether the parameter denotes the rate or mean.