Poisson process: compute E[N(3) |N(2),N(1)]

In summary, the Markov property states that the conditional probability distribution of a process at a future time only depends on the state of the process at the current time. This means that for a Poisson process with rate 1, the expected value of N(3) given N(2),N(1) is equal to the expected value of N(3).
  • #1
kingwinner
1,270
0
note: N(t) is the number of points in [0,t] and N(t1,t2] is the number of points in (t1,t2].

Let {N(t): t0} be a Poisson process of rate 1.
Evaluate E[N(3) |N(2),N(1)].

If the question were E[N(3) |N(2)], then I have some idea...
E[N(3) |N(2)]
=E[N(2)+N(2,3] |N(2)]
=E[N(2)|N(2)] + E{N(2,3] |N(2)}
=N(2)+ E{N(2,3]} (independent increments)
=N(2) + 1
since N(2,3] ~ Poisson(1(3-2)) =Poisson(1)

But for E[N(3) |N(2),N(1)], how can I deal with the extra N(1)?

Thanks for any help! :)
 
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  • #2
kingwinner said:
But for E[N(3) |N(2),N(1)], how can I deal with the extra N(1)?

Apply the Markov property
 
  • #3
bpet said:
Apply the Markov property
hmm...what is Markov property? How do we use it here?
Does it mean that E[N(3) |N(2),N(1)] = E[N(3)]?

Also, is Poisson process considered as a Markov process? Why or why not?

Can someone please explain more?

Any help is greatly appreciated!
 
  • #4
kingwinner said:
Can someone please explain more?

The wikipedia article is a good place to start.
 
  • #5
http://en.wikipedia.org/wiki/Continuous-time_Markov_process
"Markov property states that at any times s > t > 0, the conditional probability distribution of the process at time s given the whole history of the process up to and including time t, depends only on the state of the process at time t."

So according to this, it does not depend on ANY of its past history, and therefore E[N(3) |N(2),N(1)] = E[N(3)]?
 

Related to Poisson process: compute E[N(3) |N(2),N(1)]

1. What is a Poisson process?

A Poisson process is a type of stochastic process that models the occurrence of events over time. It is characterized by the assumption that the time between events follows an exponential distribution, and the number of events in a given time interval follows a Poisson distribution.

2. What does E[N(3) |N(2),N(1)] represent in a Poisson process?

E[N(3) |N(2),N(1)] represents the expected number of events that will occur in the time interval (2,3] given that N(1) events have already occurred in the time interval (0,1] and N(2) events have occurred in the time interval (1,2].

3. How is E[N(3) |N(2),N(1)] calculated?

E[N(3) |N(2),N(1)] can be calculated using the formula E[N(t) | N(s)] = N(s) + (t-s)*lambda, where lambda is the rate parameter of the Poisson process.

4. Can E[N(3) |N(2),N(1)] be negative?

No, E[N(3) |N(2),N(1)] cannot be negative. It represents the expected number of events, which is always a positive value.

5. How is the Poisson process used in real-world applications?

The Poisson process is commonly used in fields such as physics, biology, and economics to model a wide range of phenomena, including radioactive decay, neuron firing, and customer arrivals in a queue. It is also used in finance and insurance to model the occurrence of rare events, such as stock market crashes or natural disasters.

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