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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 n-1 points from 0 to t, and there is a point at exactly time t} And P= exp(-b*t)(bt)^(n-1)/(n-1)! * N/T Thanks

The words are correct, but the formula is wrong because given that N
events occur by T, the probability that n-1 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
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 n-1 points from 0 to t, and there is a point at exactly time t} And P= exp(-b*t)(bt)^(n-1)/(n-1)! * N/T Thanks
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 n-th 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
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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