Titel_SS06

 The probability of more than one event in the interval [,  is a function of a higher

order term of 2 t for 2t 0.

tt 2 t

 Events in disjoint intervals are mutually independent.

The Poisson process may be defined completely by its intensity ( ) t :

0
() lim^1
 t 2t 2 t P (one event in [,tt 2 t ) (2.51)

If ( ) t is constant in time the Poisson process is said to be homogeneous, otherwise it is
inhomogeneous.
In general the probability of n events in the interval [0, [t of a Poisson process with intensity
()t can be shown to be given as:

^0

0

()

exp ( )

!

t n

t

n

d

Pt d

n

3 3

3 3

!

"#!

$%"

$%



 # (2.52)

with mean value ENt () and variance Var N t ():

 

0

() () ( )

t

ENtVarNt 3 3d (2.53)

The probability of no events in the interval [0 i.e. is especially interesting considering

reliability problems. This probability may be determined directly from Equation

,[t Pt 0 ()

(2.52) as:

0



0

exp ( )

t

Pt 3 3d

!

"

$%

 # (2.54)

implying that the time till and between events are Exponential distributed.

From Equation (2.54) the cumulative distribution function of the waiting time till the first

event , i.e. 1 may be straightforwardly derived. Recognising that the probability of

is there is:

T 1

t

FtT() 1

)

t

T 1 & Pt 0 (



1

1 1

0

FtT 1-exp 3 3( )d

!

""

$%

 ## (2.55)

Consider now the sum of independent and exponential distributed n waiting times Tgiven as:

TTT 12 ... Tn (2.56)

It can be shown (see the lecture notes Basic Theory of Probability and Statistics in Civil

Engineering, Faber, 2006) that T is Gamma distributed:

( )(-1)exp( )

()

(1)!

n

T

ft t

n

  t



 (2.57)