1. Use the
definition of conditional probability, to show thatVirtual Laboratories > The Poisson Process > 1 2 3 4 [5] 6 7 8
Suppose that each arrival in the Poisson process, independently, is of one of two types: type 1 with probability p and type 0 with probability q = 1 - p.
This is sometimes referred to as splitting a Poisson process. For example, suppose that the arrivals are radioactive emissions and that each emitted particle is either detected (type 1) or missed (type 0) by a counter. If the arrivals are customers at a service station the each customer could be typed as either male (type 1) or female (type 0).
We are interested in the type 1 arrivals and the type 0 arrivals jointly. Let
1. Use the
definition of conditional probability, to show that
P(Mt = j, Wt = k) = P(Mt = n | Nt = j + k)P(Nt = j + k).
2. Argue that in
terms of type, the successive arrivals form a Bernoulli
trials process, and hence if there are j + k arrivals in the interval (0, t],
then the number of type 1 arrivals has the binomial
distribution with parameters j + k and p.
3. Use the results
of Exercises 1 and 2 to show that
P(Mt = j, Wt = k) = [e-rpt (rpt)j / j!][e-rqt (rqt)k / k!] for j, k = 0, 1, ...
From Exercise 3, it follows that the number of type 1 arrivals in the interval (0, t] and the number of type 2 arrivals in the interval (0, t] are independent and have Poisson distributions with parameters rpt and rqt, respectively. More generally, the type 1 arrivals and the type 2 arrivals form separate, independent Poisson processes.
4. In the
two-type Poisson experiment vary r, p, and t with
the scroll bars and note the shape of the density functions. Now set r = 2, t
= 3, and p = 0.7. Run the experiment 1000 times with an update frequency of 10 and
watch the apparent convergence of the relative frequency functions to the density
functions.
5. In the
two-type Poisson experiment, set r = 2, t = 3, and p
= 0.7. Run the experiment 500 times, updating after each run. Compute the appropriate
relative frequency functions and investigate empirically the independence of the number of
men and the number of women.
6. Suppose that
customers arrive at a service station according to the Poisson model, with rate r
= 20 per hour. Moreover, each customer, independently, is female with probability 0.6 and
male with probability 0.4. Find the probability that in a 2 hour period, there will be at
least 20 women and at least 15 men.
Suppose that Nt is not observable, but that Mt is observable. This setting is natural, for example, if the arrivals are radioactive emissions, and the type 1 arrivals are emissions that are detected by a counter while the type 0 arrivals are emissions that are missed. We would like to estimate the total number arrivals Nt in (0, t] after observing the number of type 1 arrivals Mt.
7. Show that the
conditional distribution of Nt given Mt = k
is the same as the distribution of k + Wt.
8. Show that E(Nt
| Mt = k) = k + r(1 - p)t.
Thus, if the overall rate r and the probability p that an arrival is type 1 are known, then it follows form the general theory of conditional expectation that
Mt + r(1 - p)t
is the best estimator of Nt based on Mt in the least squares sense.
9. Show that E{[Nt
- (Mt + r(1 - p)t)]2} = r(1
- p)t.
10. In the
two-type Poisson experiment, set r = 3, t = 4, and p
= 0.8. Run the experiment 100 times, updating after each run.
11. Suppose that a
piece of radioactive material emits particles according to the Poisson model at a rate of r
= 100 per second. Moreover, suppose that a counter detects each emitted particle,
independently, with probability 0.9. If the number of detected particles in a 5 second
period is 465,
Suppose that each arrival in the Poisson process, independently, is of one of k-types: type i with probability pi for i = 1, 2, ..., k. Of course we must have
p1 + p2 + ··· + pk = 1.
Let Mi(t) denote the number of type i arrivals in (0, t] for i = 1, 2, ..., k.
12. Show that for
fixed t, M1(t), M2(t),
..., Mk(t) are independent and Mi(t)
has the Poisson distribution with parameter rpit.
More generally, M1(t), M2(t), ..., Mk(t) are independent Poisson processes.