Exponential Distribution

The exponential distribution is a continuous probability distribution that describes the time between independent events occurring at a constant average rate in a Poisson process.

The exponential distribution with scale parameter $\mu >0$ has the following probability density function:

The cumulative distribution function is given by:

The distribution is often alternatively parameterized using rate parameter $\lambda =1/\mu$.

The exponential distribution finds wide application in various scientific and engineering fields:

• In reliability engineering, the exponential distribution models the time until failure of electronic components, particularly during their useful life period when failure rates are approximately constant.

• The distribution is fundamental in queueing theory, where it models both service times and inter-arrival times in Poisson processes, such as customers arriving at a service counter.

• Nuclear physicists use the exponential distribution to model the time intervals between radioactive decay events in a sample of radioactive material.

• In survival analysis and actuarial science, the exponential distribution serves as a basic model for analyzing lifetime data and mortality rates under constant hazard assumptions.

The exponential distribution has several notable properties:

• The exponential distribution exhibits the memoryless property, which means that for any times $s,t\ge 0$, the probability of waiting an additional time $t$, given that we have already waited for time $s$, is equal to the probability of waiting time $t$ from the start: $P(X>s+t|X>s)=P(X>t)$.

• The distribution has a constant hazard rate, where the hazard function $h(x)=\frac{1}{\mu }$ remains constant over time, indicating that the instantaneous probability of failure does not change with age.

• The exponential distribution possesses the maximum entropy property: among all continuous probability distributions supported on the interval $[0,\mathrm{\infty })$ with a specified mean, the exponential distribution maximizes the entropy, making it the most random or least informative distribution.

• The exponential distribution is a special case of the Gamma distribution with shape parameter $\alpha =1$

• If $X_1,\dots ,X_n$ are independent exponential random variables, their minimum follows an exponential distribution

• The exponential distribution is the continuous analog of the geometric distribution

• If $X$ follows an exponential distribution, then $Y=2X/\mu$ follows a chi-square distribution with 2 degrees of freedom

• The sum of $n$ independent exponential random variables with the same parameter follows a Gamma distribution with shape parameter $n$

• If $X$ is exponentially distributed, then $\ln (X)$ follows a Gumbel distribution

The exponential distribution is implemented by the ExponentialDistribution class. It has two constructors. The first constructor has one parameter: the scale parameter of the distribution. The following constructs an exponential distribution with waiting time (scale parameter) 7.6:

var exponential = new ExponentialDistribution(7.6);

Dim exponential = New ExponentialDistribution(7.6)

No code example is currently available or this language may not be supported.

let exponential = ExponentialDistribution(7.6)

If a variable is assumed to have an exponential distribution, then the parameter of the distribution can be estimated using the method of maximum likelihood, which gives the same results as the method of matching moments. The second constructor performs this calculation. It takes one argument: a Vector<T> whose distribution is to be estimated.

Note that parameter estimation says nothing about how well the estimated distribution fits the variable's distribution. Use one of the goodness-of-fit tests to verify the appropriateness of the choice of distribution.

The ExponentialDistribution class has one specific property, ScaleParameter, which returns the scale parameter of the distribution. This parameter commonly corresponds to the average waiting time until an event occurs.

ExponentialDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
double sample = ExponentialDistribution.Sample(random, 7.6);

Dim random = New Pcg32()
Dim sample = ExponentialDistribution.Sample(random, 7.6)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = ExponentialDistribution.Sample(random, 7.6)

The above example uses the Pcg32 to generate uniform random numbers.

For more information on the exponential distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 19. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 17. Wiley Series in Probability and Statistics.

Other Resources