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.