Exponential Distribution

The exponential distribution is the continuous counterpart to the geometric distribution. Recall that a geometric random variable counts the number of failures before the first success in a sequence of Bernoulli trials. Likewise, the random variable $x$ below will represent the waiting time until the first arrival of a success which arrives at a rate of $\lambda$ successes per unit of time.

A continuous r.v. $X$ is said to have the exponential distribution with parameter $\lambda >0$ if its PDF is

$$f(x)=\lambda e^{-\lambda x},x>0$$

We denote this by $X\sim \mathrm{Expo}(\lambda )$.

The corresponding CDF is

$$F(x)=1-e^{-\lambda x},x>0$$