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In some sense, the Poisson process is a continuous time version of the Bernoulli trials process. To see this, suppose that we think of each success in the Bernoulli trials process as a random point in discrete time. Then the Bernoulli trials process, like the Poisson process, has the strong renewal property: at each fixed time and at each arrival time, the process starts over
independently of the past. With this analogy in mind, we can see connections between three pairs of distributions.
Run the binomial experiment with and . Note the random points in discrete time.
Run the Poisson experiment with and . Note the random points in continuous time and compare with the behavior in Exercise 1.
Let us study the connection between the two processes more deeply. We will show that if we run the Bernoulli trials at a faster and faster rate but with a smaller and smaller success probability, in just the right way, the Bernoulli trials process will converge to the Poisson process. Specifically, suppose that we have a sequence of Bernoulli trials processes. In process we perform the trials at a rate of per unit time, with success probability . Our basic assumption is that as where . We will show that this sequence of Bernoulli trials processes converges, in a sense, to the Poisson process with rate parameter .
Note first that as .
Now let denote the number of successes in the time interval for Bernoulli trials process , and let denote the number of arrivals in this interval for the Poisson process. We will first show that the distribution of , which is binomial with parameters and converges to the distribution of , which is Poisson with parameter , as ,
Show the convergence of the binomial distribution to the Poisson directly, using probability density functions. That is, show that for fixed ,
Show the convergence of the binomial distribution to the Poisson using probability generating functions. That is, show that for ,
Note that the mean and variance of the binomial distribution converge to the mean and variance of the Poisson distribution, respectively.
Of course the convergence of the means is precisely our basic assumption, and is further evidence that this is the essential assumption.
More generally, show that for , the distribution of , which is binomial with parameters and , converges to the distribution of , which is Poisson with parameter , as .
Now let denote the trial number of the success for Bernoulli trials process . This variable has the negative binomial distribution with parameters and . Since the trials occur at a rate of per unit time for this process, the actual time of the success is . Let denote the time of the arrival for the Poisson process. This variable has the gamma distribution with parameters and .
Show that the distribution of converges to the distribution of as
From a practical point of view, the convergence of the binomial distribution to the Poisson means that if the number of trials
is large
and the probability of success
small
, so that the product
is of moderate size, then the binomial distribution with parameters
and
is well approximated by the Poisson distribution with parameter
. This is often a useful result, because the Poisson distribution has fewer parameters than the binomial distribution (and often in real problems, the parameters may only be known approximately). Specifically, in the limiting Poisson distribution, we do not need to know the number of trials
and the probability of success
individually, but only in the product
.
In the binomial experiment, set and , and run the simulation 1000 times with an update frequency of 10. Compute and compare each of the following:
Suppose that we have 100 memory chips, each of which is defective with probability 0.05, independently of the others. Approximate the probability that there are at least 3 defectives in the batch.
Recall that the binomial distribution can also be approximated by the normal distribution, by virtue of the central limit theorem. The normal approximation works well when and are large; the rule of thumb is that both should be at least 5. The Poisson approximation works well when is large, small so that is of moderate size.
In the binomial timeline experiment, set and and run the simulation 1000 times with an update frequency of 10. Compute and compare each of the following:
In the binomial timeline experiment, set and and run the simulation 1000 times with an update frequency of 10. Compute and compare each of the following:
A text file contains 1000 words. Assume that each word, independently of the others, is misspelled with probability .
The analogy with Bernoulli trials leads to another construction of the Poisson process. Suppose that we have a process that produces random points in time. For (measurable) , let denote the length (Lebesgue measure) of and let denote the number of random points in . Of course, is ordinary Lebesgue measure on , but note that is also a measure on our time space, albeit a random measure. Suppose that for some , the following axioms are satisfied (all sets are assumed measurable, of course):
The following exercises will show that these axioms define a Poisson process. First, let and
Use the axioms to show that satisfies the following differential equation and initial condition:
Solve the initial value problem in Exercise 14 to show that
Use the axioms to show that satisfies the following differential equation and initial condition for
Solve the differential equations in Exercise 16 recursively to show that for
From Exercise 17, it follows that has the Poisson distribution with parameter . Now let denote the arrival time for As before, we must have
Show that has the gamma distribution with shape parameter and rate parameter .
Finally, let denote the first interarrival time, and let denote the interarrival time for .
Show that the interarrival times are independent, and that each has the exponential distribution with parameter .