exponential random variable
$X$ is a exponential random variable with parameter $\lambda >0$ if its probability density function^{} is given for $x>0$ by
${f}_{X}(x)=\lambda {e}^{\lambda x}.$ 
To denote this, one usually writes $X\sim Exp(\lambda )$.
For an exponential random variable $X$:

1.
$X$ is commonly used to model lifetimes and duration between Poisson events.

2.
The expected value^{} of $X$ is given by $E[X]=\frac{1}{\lambda}$

3.
The variance^{} of $X$ is given by $Var[X]=\frac{1}{{\lambda}^{2}}$

4.
The moments of $X$ are given by ${M}_{X}(t)=\frac{\lambda}{\lambda t}$

5.
It is interesting to note that $X$ is a gamma random variable with an $\alpha $ parameter of 1.
Title  exponential random variable 

Canonical name  ExponentialRandomVariable 
Date of creation  20130322 11:54:23 
Last modified on  20130322 11:54:23 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  9 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 62E15 
Classification  msc 06F20 
Classification  msc 11B65 
Classification  msc 05C15 
Synonym  exponential distribution 