Modelling%20of%20a%20Poisson%20process

1
Modelling of a Poisson process

• Applied Statistics

2
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,
  dt) ?dt, where ? is the intensity of the
  process.

3
Random particle source

• The probability of n events happening up to time
  t is then

• This is the Poisson distribution with parameter ?t

4
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.

5
Modelling of the source

• We split the time interval into fractions of
  length dt which we can consider as
  infinitesimally small.
• In every interval (t, tdt), at most one particle
  can occur.
• P0(dt) 1 - ?dt
• P1(dt) ?dt

6
Modelling of the source

• We let the time running, and, in every interval
  dt, 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.

7
Central Limit Theorem

• The Central Limit Theorem tells us that after a
  sufficient period of time, the empirical
  distribution should approximate the theoretical
  distribution.

8
Central Limit Theorem

• I.e. the number of events in one second should
  approach the Poisson distribution with parameter
  ?.
• The time between individual events should be
  approximately exponentially distributed with
  parameter ?.