Example 2: Exponential distribution

A positive random variable is exponentially distributed $X\sim Expon(\beta)$ when $f(x)=\beta \exp {(-\beta x)}$ for $x>0$ and $\beta>0$. In other words, the random variable is more likely to take small rather than large positive values. The single parameter, $\beta$, fully determines the exponential distribution and all its moments, for example, $E(X)=1/\beta$ and $Var(X)=1/\beta^2$.