Editing Exponential distribution (section)

===Mean, variance, moments, and median===
[[File:Mean exp.svg|thumb|The mean is the probability mass centre, that is, the [[first moment]].]]
[[File:Median exp.svg|thumb|The median is the [[preimage]] ''F''<sup>−1</sup>(1/2).]]

The mean or [[expected value]] of an exponentially distributed random variable ''X'' with rate parameter ''λ'' is given by
<math display="block">\operatorname{E}[X] = \frac{1}{\lambda}.</math>

In light of the examples given [[#Occurrence and applications|below]], this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes.

The [[variance]] of ''X'' is given by
<math display="block">\operatorname{Var}[X] = \frac{1}{\lambda^2},</math>
so the [[standard deviation]] is equal to the mean.

The [[Moment (mathematics)|moments]] of ''X'', for <math>n\in\N</math> are given by
<math display="block">\operatorname{E}\left[X^n\right] = \frac{n!}{\lambda^n}.</math>

The [[central moment]]s of ''X'', for <math>n\in\N</math> are given by
<math display="block">\mu_n = \frac{!n}{\lambda^n} = \frac{n!}{\lambda^n}\sum^n_{k=0}\frac{(-1)^k}{k!}.</math>
where !''n'' is the [[subfactorial]] of ''n''

The [[median]] of ''X'' is given by
<math display="block">\operatorname{m}[X] = \frac{\ln(2)}{\lambda} < \operatorname{E}[X],</math>
where {{math|ln}} refers to the [[natural logarithm]].  Thus the [[absolute difference]] between the mean and median is
<math display="block">\left|\operatorname{E}\left[X\right] - \operatorname{m}\left[X\right]\right| = \frac{1 - \ln(2)}{\lambda} < \frac{1}{\lambda} = \operatorname{\sigma}[X],</math>

in accordance with the [[median-mean inequality]].