Transcript t - IDA

Outline of lecture 7
The Poisson process
 Definitions
 Restarted Poisson processes
 Conditioning in Poisson processes
 Thinning of point processes
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
( t ) n  t
e
(iii) P{N (t  s)  N ( s)  n} 
n!
.
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)
.
P{N (t  h)  N (t )  1}  h  o(h)
P{N (t  h)  N (t )  2}  o(h)
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/.
.
Probability theory 2011
A counting process with exponential interarrival
times is a Poisson process
P( N (t )  k )  P( N (t )  k )  P( N (t )  k  1)
 P(Tk  t )  P(Tk 1  t )  P(Tk 1  t )  P(Tk  t )


1
1 k k 1 x

k 1 x k e x dx  
 x e dx
(k  1)
( k )
t
t
k
  e  t
j 0
k
(t ) j k 1 t (t ) j
(

t
)
 e
 e  t
j!
j!
k!
j 0
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].
P{T1  s | N (1)  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].
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 N1(t) and N2(t) denote respectively the number of type I and type II events
occurring in [0, t].
Type I:
Type II:
.
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 N1(t) and N2(t) denote respectively the number of type I and
type II events occurring in [0, t].
 Then {N1(t), t  0} and {N2(t), t  0} are both Poisson processes
having respective rates p and p(1 – p). Furthermore, the two
processes are independent.
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) P{N (t  h)  N (t )  1}   (t )h  o(h)
.
P{N (t  h)  N (t )  2}  o(h)
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
N (t )
X (t )   Yi , t  0
i 1
Example: Total claims to an insurance company in the time
interval [0, t].
Expected value: ?
Variance: ?
Probability theory 2011
Exercises: Chapter VII
7.4, 7.5, 7.16, 7.18
Probability theory 2011