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rvelosoo@gmail.com science forum beginner
Joined: 09 May 2005
Posts: 16

Posted: Mon May 29, 2006 6:46 pm Post subject:
Probability of a arrival n happens at time t in Poisson



Suppose there's a birth process which is a Poisson process with
parameter b.
The birth process generates a sequence of integer numbers following a
sequence (0,1,2,3,..) during a time interval T.
Knowing that in time T the process generated N numbers, what is the
probability that number n<=N was generated precisely at time t (where t
<=T)?
Note that in the birth process, the probability that it generates a
number in a time slot delta_t is given by b*delta_t.
I expect the solution to be e^(b*t)*b*delta_t, but i'm not 100% sure.
Thanks! 

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danheyman@yahoo.com science forum beginner
Joined: 18 Jul 2005
Posts: 33

Posted: Tue May 30, 2006 12:53 am Post subject:
Re: Probability of a arrival n happens at time t in Poisson



Your guess is incorrect. Let T_n be the epoch of the nth event. Since
T_n is a cts random variable, the probability that it occurs at any
given time is zero; what you want is the density fct of the event
occuring at time t. The way to get it is to know that given that N
event have occured by time T, each of the event times has a uniform dst
on [0,T] and they're independent. Thus, the probability that exactly n
events have occured by time t has a binomial dst with "success
probability" t/T. Now you can deduce the distribution fct of T_n and
then get the density by differentiation.
Dan Heyman 

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VijaKhara@gmail.com science forum beginner
Joined: 30 Sep 2005
Posts: 26

Posted: Tue May 30, 2006 3:36 am Post subject:
Re: Probability of a arrival n happens at time t in Poisson



HOw about my solution,
The event {the point nth is at exactly time t} is equal to the event {
there are n1 points from 0 to t, and there is a point at exactly time
t}
And P= exp(b*t)(bt)^(n1)/(n1)! * N/T
Thanks 

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danheyman@yahoo.com science forum beginner
Joined: 18 Jul 2005
Posts: 33

Posted: Wed May 31, 2006 11:32 pm Post subject:
Re: Probability of a arrival n happens at time t in Poisson



VijaKhara wrote:
Quote:  HOw about my solution,
The event {the point nth is at exactly time t} is equal to the event {
there are n1 points from 0 to t, and there is a point at exactly time
t}
And P= exp(b*t)(bt)^(n1)/(n1)! * N/T
Thanks

The words are correct, but the formula is wrong because given that N
events occur by T, the probability that n1 events happened by time
t<T is not given by the Poisson dst. It is given by a binomial dst as I
described above.
Dan Heyman 

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