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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 , 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.
For , let denote -dimensional measure (technically, Lebesgue measure), defined on subsets of :
Thus, if , is the area of and if , is the volume of . Now let and consider a random process that produces random points in . For , let denote the number of random points in . This collection of random variables is a Poisson process on with density parameter if the following axioms are satisfied:
By convention, if then with probability 1, and if then with probability 1. On the other hand, note that if then with probability 1.
In the two-dimensional Poisson process, vary the width and the rate . Note the location and shape of the density of . Now with and , 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 of the Poisson distribution, show that for ,
In particular, can be interpreted as the expected density of the random points, justifying the name of the parameter
In the two-dimensional Poisson process, vary the width and the density parameter . Note the size and location of the mean-standard deviation bar of . Now with and , run the simulation 1000 times with an update frequency of 10. Note the apparent convergence of the empirical moments to the true moments.
Suppose that defects in a sheet of material follow the Poisson model with an average of 1 defect per 2 square meters. Consider a 5 square meter sheet of material.
Suppose that raisins in a cake follow the Poisson model with an average of 2 raisins per cubic inch. Consider a slab of cake that measures 3 by 4 by 1 inches.
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 with density parameter . For , let where
is the circular region of radius , centered at the origin. Let and for let denote the distance of the closest point to the origin. Note that is the analogue of the arrival time for the Poisson process on .
Show that has the Poisson distribution with parameter .
Show that if and only if .
Show that has the gamma distribution with shape parameter and rate parameter .
Show that has probability density function
Show that for are independent and each has the exponential distribution with rate parameter .
Again, the Poisson model defines the most random way to distribute points in space, in a certain sense. Specifically, consider the Poisson process on with parameter .
Suppose that contains exactly one random point. Show that the position of the point is uniformly distributed in .
More generally, if contains points, then the positions of the points are independent and each is uniformly distributed in .
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 meter.
Suppose that and . Show that the conditional distribution of given is the binomial distribution with trial parameter and success parameter
.More generally, suppose that and that is partitioned into subsets . Show that the conditional distribution of given is the multinomial distribution with parameters and , where for each ,
.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.
Suppose that is a Poisson process in with density parameter . Splitting this Poisson process works just like splitting of the standard Poisson process. Specifically, suppose that the random points are of different types and that each random point, independently of the others, is type with probability . Let denote the number of type points in a region , for . Of course, we must have
Show that for ,
More generally, is a Poisson processes with density parameter for each , and these processes are independent.
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.
We can simulate a Poisson variable using the general quantile method.
Suppose that is probability density function on . If is uniformly distributed on (a random number), show that variable defined below has probability density function :
Now we can use the result in the previous exercise to simulate a Poisson process in a region . We will illustrate the method with the rectangle . where and . First, we use a random number to simulate a random variable that has the Poisson distribution with parameter . Next, if , let and be sequences of random numbers, and define
Show that the random points of the Poisson process with rate on are simulated by for .