#### Motivation

If you have understood how the Poisson process works and are willing to accept that a Gamma distribution models the time to wait to observe *a* events, this section is superfluous to your needs. We explain here:

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How the Exponential distribution models the time to wait for the first event and arises naturally out of a memoryless system;

And therefore why the distribution of the time to wait to observe an event remains the same even if one has waited a while;

How the Gamma distribution is the sum of a number of Exponential distributions, and thus is the waiting time distribution for

*a*events.

#### Deriving the Exponential distribution

The Poisson process assumes that there is a constant probability that an event will occur per increment of time. If we consider a small element of time * Dt*, then the probability an event will occur in that element of time is

*k*, where

*D*t*k*is some constant. Now let

*P(t)*be the probability that the event will not have occurred by time

*t*. The probability that an event occurs the first time during the small interval

*after time*

*D*t*t*is then

*k*. This is also equal to

*D*tP(t)*P(t) - P(t+*and we have:

*D*t)...

which is the cumulative distribution function for an Exponential distribution Exponential(*k*) with mean 1/*k*. Thus 1/*k* is the mean time between occurrences of events or, equivalently, *k* is the mean number of events per unit time, which is the Poisson parameter * l*. The parameter 1/

*, the mean time between occurrences of events, is given the notation*

*l**.*

*b*

#### Derivation of the Gamma distribution

We have shown above that the time until occurrence of the first event for a Poisson distribution is given by:

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LaTeX Math Block | ||
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Gamma(0,\beta,a)=\displaystyle\sum_{a}Exponential(1/\beta) |

#### The memoryless property of an Exponential distribution

The probability that the first event will occur at time *x*, given it has not yet occurred by time *t* (*x*>*t*), is given by:

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