Exponential(1/b)

Crystal Ball parameter restrictions

The Exponential(1/*b*) is a right-skewed distribution bounded at zero with a mean of b. It only has one shape. Examples of the Exponential distribution are given below:

#### Uses

The Exponential(1/*b*) models the time until the occurrence of a first event in a Poisson process. For example:

The time until the next earthquake;

The decay of a particle in a mass of radioactive material;

The length of telephone conversations.

The parameter *b* is the mean time until the occurrence of the next event.

#### Example

An electronic circuit could be considered to have a constant instantaneous failure rate, meaning that at a small interval of time it has the same probability of failing, given it has survived so far. Destructive tests show that the circuit lasts on average 5,200 hours of operation. The time until failure of any single circuit can be modeled as Exponential (1/5200) hours. Interestingly, if it conforms to a true Poisson process, this estimate will be independent of how many hours of operation, if any, the circuit has already survived.

#### Comments

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.

The Exponential distribution is sometimes called the *Negative Exponential*. The Exponential distribution is a special case of the Weibull distribution: Weibull(*0,b,1*)= Exponential(*1/**b*).

The Excel function EXPONDIST(x,1/b,**0**) returns the Exponential probability density function, and the EXPONDIST(x,1/b,**1**) returns the Exponential cumulative distribution function.

Note: be careful about the 1/b.