1. Probability theory 2011 Outline of lecture 7 The Poisson process  Definitions  Restarted Poisson processes  Conditioning in Poisson processes  Thinning of point processes

2. Probability theory 2011 The Poisson process  The counting process {N(t), t  0} is said to be a Poisson process having rate > 0, if (i) N(0) = 0 (ii) The number of events in disjoint intervals are independent (iii).

3. Probability theory 2011 The Poisson process – another definition  The counting process {N(t), t  0} is a Poisson process having rate, > 0, if and only if (i) N(0) = 0 (ii) The number of events in disjoint intervals are independent (iii).

4. Probability theory 2011 The Poisson process – yet another definition  The counting process {N(t), t  0} is a Poisson process having rate, > 0, if and only if (i) N(0) = 0 (ii) The interarrival times are independent identically distributed exponential random variables having mean 1/..

5. Probability theory 2011 A counting process with exponential interarrival times is a Poisson process

6. Probability theory 2011 Conditional distribution of arrival times  Given that N(1) = 1, the arrival time of the first event is uniformly distributed on [0, 1].  Given that N(1) = n, the arrival times have the same distribution as the order statistics corresponding to n independent variables uniformly distributed on [0, 1].

7. Probability theory 2011 Thinning of Poisson processes  Consider a Poisson process {N(t), t  0} having rate, > 0.  Classify each event as a type I event with probability p and a type II event with probability 1-p independently of the other events.  Let N 1 (t) and N 2 (t) denote respectively the number of type I and type II events occurring in [0, t]. Type I: Type II:.

8. Probability theory 2011 Thinning of Poisson processes  Consider a Poisson process {N(t), t  0} having rate, > 0.  Classify each event as a type I event with probability p and a type II event with probability 1-p independently of the other events.  Let N 1 (t) and N 2 (t) denote respectively the number of type I and type II events occurring in [0, t].  Then {N 1 (t), t  0} and {N 2 (t), t  0} are both Poisson processes having respective rates p and p(1 – p). Furthermore, the two processes are independent.

9. Probability theory 2011 Nonhomogeneous Poisson processes  The counting process {N(t), t  0} is said to be nonhomogeneous Poisson process with intensity function (t), t  0, if (i) N(0) = 0 (ii) The number of events in disjoint intervals are independent (iii).

10. Probability theory 2011 Compound Poisson processes  A stochastic process {X(t), t  0} is said to be a compound Poisson process if it can be represented as Example: Total claims to an insurance company in the time interval [0, t]. Expected value: ? Variance: ?

11. Probability theory 2011 Exercises: Chapter VII 7.4, 7.5, 7.16, 7.18