Math 413                                    Poisson Process (1)                            Winter 2002

                                                

 

Counting Process

 

A stochastic process {is a counting process if N(t) represents the total number of “events” that have occurred up to (and including) time t.

 

Example1:

a)  N(t) = the number of persons who have entered a particular store at or prior to time t.

 

b)  N(t) = the number of infants who were born by time t.

 

c)  N(t) = the number of goals that a given soccer player has scored by time t.

 

 

Basic Properties of a Counting Process

 

  1. N(t) ³ 0.
  2. N(t) is integer valued.
  3. If s < t, then N(s) £ N(t).
  4. For s < t, N(t) - N(s) equals the number of times that the event has occurred in the time interval (s, t].

 

 

Independent Increment

 

A counting process is said to possess independent increments if the numbers of events that occur in disjoint time intervals are independent.

 

Example2:

If N(t) possesses independent increments, then N(10) is independent of N(15)-N(10).

Number of events that occur by time 10 must be independent of the number of events

that occur between times 10 and 15.

 

 

Stationary Increment

 

A counting process is said to possess stationary increments if the distribution of number of events that occur in any time interval depends ONLY on the LENGTH of the time interval. [Note: Not depend on the timing of the time interval]

 

 

 

 

 

Example3:

If the process N(t) has stationary increments, then N(t1+s) - N(t2+s)  has the same distribution as N(t2) -N(t1) for all .

 

 

 

Discuss Example1 (a), (b), and (c). Which possess independent increments and which

stationary increments?

 

 

 

 

Poisson Process

 

The counting process {is said to be a Poisson Process having rate l, l>0, if

  1. N(0) = 0
  2. has independent increments
  3. The number of events in any interval of length t is Poisson distributed with mean lt.  That is,

                               If X = N(t + s) - N(s), then P(X = x) =  , X = 0, 1, ….

 

       According to 3., a Poisson Process has stationary increments and .

 

 

 

 

Conditional Distribution of the Arrival Times

 

Suppose we are told that exactly one event of a Poisson process has taken place by time t, and we want to determine the distribution of the time at which the event occurred. Now since a Poisson process possesses stationary and independent increments if seems reasonable that each interval in [0,t] of equal length should have the same probability of containing the event. In other words, the time of event should be uniformly distributed over [0,t].

 

Prove the above argument by calculating the conditional probability P[ T< s | N(t) = 1] .

 

 

 

 

 

 

 

Math 413                                    Poisson Process (2)                           Winter 2002

 

 

Elapsed Time –another way of defining a Poisson process

 

Let denote the elapsed time between the (n-1)st and the nth event. The sequence {, n=1, 2, …}is called the sequence of interarrival times.

 

, n=1, 2, … are independent identically distributed exponential random variables having mean . We define a Poisson (counting) process by saying that the nth event occurs at time  .

 

 

 

1. Suppose that ducks immigrate into Yakima River at a Poisson rate = 2 per week.

a)      What is the expected time until the tenth duck arrives?

 

 

 

 

 

 

b)      What is the probability that the elapsed time between the tenth and the eleventh arrival exceeds two weeks? 

 

 

 

 

 

 

 

Poisson Process-Two Types of Events involved

If

 

Then {, t ³ 0} is a Poisson process having rate .

 

         {, t ³ 0} is a Poisson process having rate .

 

          And the two processes are independent!

2. If immigrants to Washington arrive at a Poisson rate of 10 per week, and if each immigrant is of English descent with probability 1/12, then what is the probability that no people of English descent will emigrate to Washington state during the month of February?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Math 413                                    Poisson Process (3)                           Winter 2002

 

 

                                                Compound Poisson Process

 

A stochastic process {} is a compound Poisson process if  where

 

{} is a Poisson process, and {} is a family of independent and identically distributed random variables which are also independent of {}.

 

 

E[] = E[E[|] = E[] E[] =E[]

 

Var[] = E[] Var[] + Var[] (E[])2

 

                           = [ Var[]+(E[])2 ]

 

                  =  E[2]

 

 

 

When is discrete, as t grows large, the distribution ofconverges to a normal distribution. Why??

 

An event is called TYPE j if Y=;  = # of TYPE j events by time t.

Express =        

Based on the theorem you have learned, , , are independent Poisson random variables with respective means .

 

Math 411:  Poisson() Normal()        as

Thus, Poisson() Normal()        as

 

 

 

 

 

 

 

 

1. Suppose that families move to Leavenworth at a Poisson rate = 2 per month. If the number of people in each family is independent and takes on the values 1, 2, 3, 4 with respective probabilities 1/6, 1/3, 1/3, 1/6, then what is the expected value and variance of the number of individuals moving to Leavenworth during one-year period.

 

 

 

 

 

 

 

 

 

 

2. Exercise 77 & 80 on page 309

 

 

 

 

 

 

 

 

 

 

 

 

 

3. Continued from 1., find the approximate probability that at least 240 people move to Leavenworth within the next 50 months.