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Probability of a arrival n happens at time t in Poisson
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danheyman@yahoo.com
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Joined: 18 Jul 2005
Posts: 33

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

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
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VijaKhara@gmail.com
science forum beginner


Joined: 30 Sep 2005
Posts: 26

PostPosted: Tue May 30, 2006 3:36 am    Post subject: Re: Probability of a arrival n happens at time t in Poisson Reply with 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
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danheyman@yahoo.com
science forum beginner


Joined: 18 Jul 2005
Posts: 33

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

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
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rvelosoo@gmail.com
science forum beginner


Joined: 09 May 2005
Posts: 16

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

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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