Virtual Laboratories > The Poisson Process > 1 2 3 4 5 6 [7] 8
The Poisson process can be defined in higher dimensions, as a model of random points in space. Some specific examples of "random points" are
Our original construction of the Poisson process on [0, 
), starting with the interarrival
times, does not generalize easily, because this construction depends critically on the
order of the real numbers. However, the alternate construction in the last
section, motivated by the analogy with Bernoulli trials,
generalizes very naturally.
Fix k, let m denote k-dimensional measure, defined on
subsets of Rk. Thus, if k = 2, m(A)
is the area of A and if k = 3, m(A) is the
volume
of A. Now let D be a subset of Rk
and consider a random process that produces random points in D. For A 
D with m(A) positive and
finite, let N(A) denote the number of random points in A. This
collection of random variables is a Poisson process on D with density
parameter r if the following axioms are satisfied:
1. In the
two-dimensional Poisson
process, vary the width w and the rate r.
Note the location and shape of the density of N. Now with w = 3 and r
= 2, run the simulation 1000 times with an update frequency of 10. Note the apparent
convergence of the empirical density to the true density.
Using our previous results on moments, it follows that
E[N(A)] = r m(A), var[N(A)] = r m(A).
In particular, r can be interpreted as the expected density of the random points, justifying the name of the parameter
2. In the
two-dimensional Poisson
process, vary the width w and the
density parameter r. Note the size and location of the mean-standard deviation
bar of N. Now with w = 4 and r = 3, run the simulation 1000
times with an update frequency of 10. Note the apparent convergence of the empirical
moments to the true moments.
3. Suppose that
defects in a sheet of material follow the Poisson model with an average of 1 defect per 2
square meters. In a 5 square meter sheet of material,
4. Suppose that
raisins in a cake follow the Poisson model with an average of 2 raisins per
cubic inch.
In a slab of cake that measures 3 by 4 by 1 inches,
5. Suppose that
the occurrence of trees in a forest of a certain type that exceed a certain critical size
follows the Poisson model. In a one-half square mile region of the forest there are 40
trees that exceed the specified size.
Consider the Poisson process in R2 with density parameter r.
For t > 0, let Mt = N(Ct)
where Ct is the circular region of radius t, centered at the
origin. Let Z0 = 0 and for k = 1, 2, ... let Zk
denote the distance of the k'th closest point to the origin. Note that Zk
is the analogue of the k'th arrival time for the Poisson process on [0, ).
6. Show that Mt
has the Poisson distribution with parameter 
t2r.
7. Show that Zk
t if and only if Mt 
k.
8. Show that
Zk2 has the gamma distribution
with shape parameter k and rate parameter r.
9. Show that Zk
has density function
g(z) = 2( r)k
z2k - 1 exp(- r z2)
/ (k - 1)!, z > 0.
10. Show that
Zk2 -
Zk - 12, k = 1, 2, ... are independent
and each has the exponential distribution with rate parameter r.
Again, the Poisson model defines the most random way to distribute points in space, in a certain sense. Specifically, consider the Poisson process on Rk with parameter r. Recall again that we consider subsets A of Rk with m(A) positive and finite.
11. Suppose that a
region A contains exactly one random point. Show that the position
More generally, if A contains n points, then the positions of the points are independent and each is uniformly distributed in A.
12. Suppose
that
defects in a type of material follow the Poisson model. It is known that a square sheet
with side length 2 meters contains one defect. Find the probability that the defect is in
a circular region of the material with radius 1/4 meter.
13. Suppose that a
region A has n random points. Let B be a subset of A.
Show that the number of random points in B has the binomial distribution with parameters n
and p = m(B) / m(A).
14. More
generally, suppose that a region A is partitioned into k subsets B1,
B2, ..., Bk. Show that the conditional
distribution of (N(B1), N(B2),
..., N(Bk)) given N(A) = n is multinomial with parameters n and pi
= m(Bi) / m(A), i = 1, 2, ..., k.
15. Suppose that
raisins in a cake follow the Poisson model. A 6 cubic inch piece of the cake with 20
raisins is divided into 3 equal parts. Find the probability that each piece has at least 6
raisins.
Splitting of a k-dimensional Poisson process works just like splitting of the standard Poisson process. Specifically, suppose that the random points are of j different types and that each random point, independently of the others is type i with probability pi for i = 1, 2, ..., j. Let Ni(A) denote the number of type i points in a region A, for i = 1, 2, ..., j.
16. Show that
More generally, the points of type i form a Poisson processes with density parameter rpi for each i, and these processes are independent.
17. Suppose that
defects in a sheet of material follow the Poisson model, with an average of 5 defects per
square meter. Each defect, independently of the others is mild with probability
0.5, moderate with probability 0.3, or severe with probability 0.2.
Consider a
circular piece of the material with radius 1 meter.