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.

## Exponential Distribution Mean

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

### Chi square Distribution

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