The Exponential Distribution

Exponential distribution

Probability Density Function

Let $\lambda>0$

\begin{equation*} f(x;\lambda)=\lambda e^{-\lambda x} \end{equation*}

For $x\ge0$.

This is the special case of the gamma function with $\alpha=1,\beta=\frac{1}{\lambda}$.

The cdf:

\begin{equation*} F(x;\lambda)=1-e^{-\lambda x},x\ge0$ \end{equation*}

The $p$ th percentile: $x=-\frac{1}{\lambda}\ln\left(1-p\right)$

Mean and Variance

$\mu=\frac{1}{\lambda}$

$\sigma=\frac{1}{\lambda}$

Relation to the Poisson Distribution

It is commonly used for the distribution of the times between successive events (similar to the Poisson distribution).

If the number of events in any time interval of length $t$ is Poisson with $\lambda=\alpha t$, and the number of occurrences in nonoverlapping intervals is independent, then the distribution of elapsed time between two successive events is exponential with $\lambda=\alpha$.

It is the continuous analog of the geometric distribution.

Properties

A nice property is that it is memoryless. The probability of failure in time $t+t_{0}$ given it survived to time $t_{0}$ is the same as the probability of it surviving to time $t$.

It is one of the only two distributions that are memoryless.