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Q3. Let N(t) : t ? 0 be a Poisson process (N(0) = 0). The Poisson process is an independent increment process. The two key properties of the process are: Property 1 : Consider any interval [t1, t2] such that t2 ? t1. Then, the number of arrivals in interval [t1, t2] follows Poisson distribution with parameter ?(t2 ? t1), i.e., N(t2) ? N(t1) ? Poisson(?(t2 ? t1)) Property 2 : Consider any two non overlapping intervals, i.e., for any t1 < t2 < t3 < t4, [t2 ? t1]?[t4 ? t3] = ?. Then, N(t2) ? N(t1) and N(t4) ? N(t3) are independent. Show that for s = t/6, the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = 1/6, i.e., P(N(s) = m|N(t) = n) = (n over m)(1 ? p)^{n?m}p^m.

          Q3. Let N(t) : t ? 0 be a Poisson process (N(0) = 0). The Poisson process is an independent increment process. The two key properties of the process are:

Property 1 : Consider any interval [t1, t2] such that t2 ? t1. Then, the number of arrivals in interval [t1, t2] follows Poisson distribution with parameter ?(t2 ? t1), i.e., N(t2) ? N(t1) ? Poisson(?(t2 ? t1))

Property 2 : Consider any two non overlapping intervals, i.e., for any t1 < t2 < t3 < t4, [t2 ? t1]?[t4 ? t3] = ?. Then, N(t2) ? N(t1) and N(t4) ? N(t3) are independent.

Show that for s = t/6, the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = 1/6, i.e.,

P(N(s) = m|N(t) = n) = (n over m)(1 ? p)^{n?m}p^m.

Q3. Let N(t) : t ? 0 be a Poisson process (N(0) = 0). The Poisson process is an independent increment process. The two key properties of the process are:

Property 1 : Consider any interval [t1, t2] such that t2 ? t1. Then, the number of arrivals in interval [t1, t2] follows Poisson distribution with parameter ?(t2 ? t1), i.e., N(t2) ? N(t1) ? Poisson(?(t2 ? t1))

Property 2 : Consider any two non overlapping intervals, i.e., for any t1 < t2 < t3 < t4, [t2 ? t1]?[t4 ? t3] = ?. Then, N(t2) ? N(t1) and N(t4) ? N(t3) are independent.

Show that for s = t/6, the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = 1/6, i.e.,

P(N(s) = m|N(t) = n) = (n over m)(1 ? p)^n?mp^m.

Added by David H.

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