Exponential distribution

Exponential distribution pdf - public domain

Exponential distribution cdf - public domain

Mean exp

Median exp

Tukey anomaly criteria for Exponential PDF

Exponential distribution is a probability distribution used in statistics to model the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution and is related to various other statistical distributions and concepts.

Definition

The exponential distribution is defined by its probability density function (PDF):

\[ f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases} \]

where \( \lambda \) is the rate parameter of the distribution, often called the inverse scale parameter, and \( e \) is the base of the natural logarithm. The parameter \( \lambda \) must be greater than zero. The exponential distribution is often concerned with the amount of time until some specific event occurs for the first time.

Properties

The exponential distribution has several key properties:

• Memorylessness: The exponential distribution is the only continuous distribution that has the memoryless property, meaning that the probability of an event occurring in the future is independent of how much time has already elapsed.

• Mean and Variance: The mean (or expected value) of an exponential distribution is \( \frac{1}{\lambda} \), and the variance is \( \frac{1}{\lambda^2} \).

• Exponential Family: It is a member of the exponential family of distributions.

• Relation to Poisson Process: The exponential distribution is closely related to the Poisson distribution. In a Poisson process, the time between events follows an exponential distribution.

Applications

The exponential distribution is widely used in various fields such as:

• Queueing theory, to model the times between arrivals of customers at a service point.
• Reliability engineering and failure time analysis, to model the time until a system or component fails.
• Biology and medicine, to model the time until an event such as the death of an organism or the time between occurrences of infectious diseases.

Parameter Estimation

The parameter \( \lambda \) of the exponential distribution can be estimated using the maximum likelihood estimation (MLE) method. Given a sample of \( n \) independent and identically distributed observations from an exponential distribution, the MLE of \( \lambda \) is given by:

\[ \hat{\lambda} = \frac{n}{\sum_{i=1}^{n} x_i} \]

where \( x_i \) are the observed values.

Related Distributions

• If \( X \) is exponentially distributed with rate \( \lambda \), then \( kX \) for \( k > 0 \) is also exponentially distributed but with rate \( \frac{\lambda}{k} \).
• The exponential distribution is a special case of the gamma distribution with shape parameter \( k=1 \).
• The sum of \( n \) independent exponential (with the same rate parameter \( \lambda \)) random variables follows a gamma distribution with parameters \( k=n \) and \( \theta=\frac{1}{\lambda} \).