# exponential random variable

$X$ is a exponential random variable with parameter $\lambda>0$ if its probability density function is given for $x>0$ by

 $\displaystyle f_{X}(x)=\lambda e^{-\lambda x}.$

To denote this, one usually writes $X\sim Exp(\lambda)$.

For an exponential random variable $X$:

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

The moments of $X$ are given by $M_{X}(t)=\frac{\lambda}{\lambda-t}$

5. 5.

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

Title exponential random variable ExponentialRandomVariable 2013-03-22 11:54:23 2013-03-22 11:54:23 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 62E15 msc 06F20 msc 11B65 msc 05C15 exponential distribution