Editing Exponential distribution (section)

==Definitions==

===Probability density function===
The [[probability density function]] (pdf) of an exponential distribution is

:<math> f(x;\lambda) = \begin{cases}
\lambda  e^{ - \lambda x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>

Here ''λ'' > 0 is the parameter of the distribution, often called the ''rate parameter''. The distribution is supported on the interval&nbsp;{{closed-open|0, ∞}}. If a [[random variable]] ''X'' has this distribution, we write&nbsp;{{math|''X'' ~ Exp(''λ'')}}.

The exponential distribution exhibits [[infinite divisibility (probability)|infinite divisibility]].

===Cumulative distribution function===
The [[cumulative distribution function]] is given by

:<math>F(x;\lambda) = \begin{cases}
1-e^{-\lambda x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>

===Alternative parametrization===
The exponential distribution is sometimes parametrized in terms of the [[scale parameter]] {{math|1=''β'' = 1/''λ''}}, which is also the mean:
<math display="block">f(x;\beta) = \begin{cases}
   \frac{1}{\beta} e^{-x/\beta} & x \ge 0, \\
    0 & x < 0.
 \end{cases} \qquad\qquad
F(x;\beta) = \begin{cases}
   1- e^{-x/\beta} & x \ge 0, \\
    0 & x < 0.
 \end{cases} </math>