Exponential distribution can be viewed as the distribution of waiting time between the occurrence of events described by a Poisson variable. If T is the time elapsed before the occurrence of a Poisson event, it can also be shown that T is the recurrence time between the occurrence of any two events. This analysis will show that the distribution of the variable T is the exponential.
Definition: If X is a continuous random variable with pdf
f(x) = $\lambda e^{-\lambda x}$ for x > 0
= 0 otherwise
then X is said to have an exponential distribution with parameter λ.
The cdf of an exponential variable is F(x) = 1- $e^{-\lambda x}$
The parameter λ of the exponential distribution represents the average number of events per unit time in the corresponding Poisson Process.

Negative exponential distribution is the other name by which exponential distributions are known.

The mean or the expected value of an exponential distribution is given in terms of the parameter λ of the distribution.

μ = E(X) = $\frac{1}{\lambda }$

Sometimes an exponential distribution is also described using μ, the mean as the parameter as follows

f(x) = $\frac{1}{\mu }e^{-\frac{x}{\mu }}$