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```Modelling of a Poisson
process
Applied Statistics
Random particle source
Conditions
 Probability of an event occurring in an
infinitesimal interval does not depend
on previous events.
 Events can only occur singly.
Then the probability of an event occurring
in (t, δt) ~ λ·δt, where λ is the intensity
of the process.
Random particle source

The probability of n events happening
up to time t is then
Pn  t  

t 
n
e
 t
n!
This is the Poisson distribution with
parameter λt
Time between events
We are not only interested in the
number of events during a time interval
but also in the distribution of time
between individual events.
 Time between events is a random
variable with exponential distribution,
i.e.
t
fT  t    e
t 0

Modelling of the source
We split the time interval into fractions
of length δt which we can consider as
infinitesimally small.
 In every interval (t, t+δt), at most one
particle can occur.
P0(δt) = 1 - λ·δt
P1(δt) = λ·δt

Modelling of the source
We let the time running, and, in every
interval δt, we randomly decide whether
the event occurs or not.
 We store the times between these
events.
 After every second, we count the
number of events in the passed second.

Central Limit Theorem

The Central Limit Theorem tells us that
after a sufficient period of time, the
empirical distribution should
approximate the theoretical distribution.
Central Limit Theorem
I.e. the number of events in one second
should approach the Poisson
distribution with parameter λ.
 The time between individual events
should be approximately exponentially
distributed with parameter λ.

```
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