Exponential distribution | Vose Software

# Exponential distribution

Format: Expon(b)

The Expon(b) is a right-skewed distribution bounded at zero with a mean of b. It only has the one shape. Examples of the Exponential distribution are given below:

## Uses

If risk events are assumed to occur randomly in time (i.e. follow a Poisson process) and the average time between events equals β, then the time between each consecutive event will be distributed according to Expon(β). So, for example, if an insurer sees that some particular type of natural disaster occurs on average once every 5.5 years, the time between such consecutive disasters can be modeled as Expon(1/5.5) years.

The memoryless property of the Exponential distribution also means that the time until the next event (even though it may have been some time since the last such event occurred) also follows an Expon distribution. This implies a rather aesthetically pleasing property: the right tail of an Exponential distribution takes the same shape as its whole.

One method of testing whether events are occurring randomly in time is to test whether an Exponential distribution fits well to the times between each event.

The Exponential and Geometric distributions are the only distributions that allow for independence between additional waiting time and elapsed waiting time (sometimes described as a process that has no memory). When a Poisson distribution is a good approximation to a Binomial, an Exponential distribution is also a good approximation to a Geometric.

If events occur randomly, the Exponential distribution shows that the next event is more likely to occur immediately than at any other time. That may seem strange, but for the next event to occur at a later moment it must not have occurred in any previous moment, and since it is just as likely to occur at any moment, 'now' has no conditions whereas 'later' does.

The Exponential distribution is sometimes called the Negative Exponential distribution. The Exponential distribution is a special case of the Weibull distribution: Weibull(1, b) = Expon(b).

## ModelRisk functions added to Microsoft Excel for the Exponential distribution

VoseExponential generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseExponentialObject constructs a distribution object for this distribution.

VoseExponentialProb returns the probability density or cumulative distribution function for this distribution.

VoseExponentialProb10 returns the log10 of the probability density or cumulative distribution function.

VoseExponentialFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseExponantialFitObject constructs a distribution object of this distribution fitted to data.

VoseExponentialFitP returns the parameters of this distribution fitted to data.