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.