Exponential distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A continuous distribution of a random variable $ X $ defined by the density

\begin{equation} \label{eq1} p(x) = \begin{cases} \lambda e^{-\lambda x}, \quad x \geq 0 , \\ 0, \quad x < 0. \end{cases} \end{equation}

The density $ p( x) $ is dependent on the positive scale parameter $ \lambda $. The formula for the moments is $ {\mathsf E} X ^ {n} = n!/ \lambda ^ {n} $, and, in particular, the expectation equals $ {\mathsf E} X = 1/ \lambda $ and the variance equals $ {\mathsf D} X = 1/ \lambda ^ {2} $; the characteristic function is $ ( 1- it/ \lambda ) ^ {-1} $.

The exponential distribution belongs to the family of gamma-distributions (cf. Gamma-distribution) which are defined by the densities

$$ p( x) = \frac{\lambda ^ \alpha x ^ {\alpha - 1 } }{\Gamma ( \alpha ) } e ^ {- \lambda x } ,\ \ x \geq 0,\ \alpha > 0 ; $$

the $ n $- fold convolution of the density (1) is equal to the gamma-density with the same parameter $ \lambda $ and with $ \alpha = n $.

The exponential distribution is the unique distribution having the property of no after-effect: For any $ x > 0 $, $ y > 0 $ one has

\begin{equation} \label{eq2} {\mathsf P} \{ X > x + y \mid X > y \} = \ {\mathsf P} \{ X > x \} , \end{equation}

where $ {\mathsf P} \{ X > x + y \mid X > y \} $ is the conditional probability of the event $ X > x + y $ subject to the condition $ X > y $. Property (2) is also called the lack-of-memory property.

In a homogeneous Poisson process, the distances between successive events have an exponential distribution. Conversely, a renewal process with exponential lifetime \eqref{eq1} is a Poisson process. An exponential distribution often arises as a limit process on the superposition or extension of renewal processes, as well as in high-level intersection problems in various random-path schemes, in critical branching processes, etc.

The above features explain why the exponential distribution is widely used in calculating various systems in queueing theory and reliability theory. One assumes that the lifetimes of the devices are independent random variables with exponential distributions, and then the property \eqref{eq2} enables one to examine a queueing system by means of finite or denumerable Markov chains with continuous time. Similarly, one uses Markov chains in reliability theory, where the fault-free operating times of the individual devices can often be taken as independent and as having exponential distributions.

Comments

The lack-of-memory property is related to the Markov property in Poisson processes.

References

[F]	W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971)
[R]	S.M. Ross, "Stochastic processes" , Wiley (1983) MR0683455 Zbl 0555.60002 Zbl 0568.60079
[S]	A.D. Solovyev, "Mathematical methods of reliability theory" , Acad. Press (1969) (Translated from Russian) MR0345234 Zbl 0169.50501