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title: "Time to wait to observe alpha events"
canonical: "https://modelassist.epixanalytics.com/space/EA/26574981/Time%20to%20wait%20to%20observe%20alpha%20events"
format: markdown
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####
#### 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 *α* events, this section is superfluous to your needs. We explain here:
- How the [Exponential distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/) 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/) is the sum of a number of Exponential distributions, and thus is the waiting time distribution for *α* 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 *kDt*, where *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 *Dt* after time *t* is then *kDtP(t)*. This is also equal to *P(t) - P(t+Dt)* and we have:
> Macro (mathblock)
Making *Dt* infinitesimally small, this becomes the differential equation:
> Macro (mathblock)
Integration gives:
> Macro (mathblock)
> Macro (mathblock)
If we define *F(t)* as the probability that the event will have occurred before time *t* (i.e. (1-*P(t)*), the cumulative distribution function for *t*), we then have:
> Macro (mathblock)
which is the cumulative distribution function for an [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/) 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 *λ*. The parameter 1/*λ*, the mean time between occurrences of events, is given the notation β.
#### 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:
*t*1 = Exponential(1/β) where β = 1/*λ*
From the mathematics of convolutions we have:
> Macro (mathblock)
For X = [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575671/)(0,β,*α*) and Y = [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575672/)(1/β) this gives:
> Macro (mathblock)
This is equal to a Gamma (0,β,*α* +1).
Since Gamma(0,β*,*1) = Y = [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575672/)(1/β) we have proven by induction that:
> Macro (mathblock)
#### 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:
> Macro (mathblock)
which is another Exponential distribution. Thus, although the event may not have occurred after time *t*, the remaining time until it will occur has the same probability distribution as it had at any prior point in time.
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