1. Run the
binomial experiment with n = 50 and p = 0.1. Note the random points in
discrete time.Virtual Laboratories > The Poisson Process > 1 2 3 4 5 [6] 7 8
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 regenerative 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.
1. Run the
binomial experiment with n = 50 and p = 0.1. Note the random points in
discrete time.
2. Run the
Poisson experiment with t = 5 and r = 1. Note the random points in
continuous time and compare with the behavior in Exercise 1.
Let us study the connection between the binomial and Poisson distributions more deeply. Consider the binomial distribution in which the success parameter p depends on the number of trials n. Moreover, suppose that
npn 
c as n
.
3. Show that this
assumption implies that
pn 0 as n .
so that the probability of success is small when the number of trials is large.
We will show that this binomial distribution converges, as n increases, to the Poisson distribution with parameter c.
4. For a fixed
nonnegative integer k, show that
C(n, k) pnk (1 - pn)n - k = (1 / k!)npn(n - 1)pn ··· (n - k + 1)pn (1 - npn / n)n - k.
The left side of the equation in Exercise 4 is the binomial probability density function evaluated at k.
5. Show that for
fixed j,
(n - j)pn c as n .
6. Use a
theorem from calculus to show that fixed k,
(1 - npn / n)n-k e-c as n .
7. Use the
results of Exercises 4-6 to show that
C(n, k) pnk (1 - pn)n
- k e-c
ck / k! as n .
8. In the
binomial experiment, set n = 30 and p = 0.1 and run the simulation 1000
times with an update frequency of 10. Compute and compare each of the following:
4) 4}. 4) 9. In the setting
of this section, show that the mean and variance of the binomial distribution converge to
the mean and variance of the Poisson distribution, respectively, as n increases.
10. 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 be approximated by the normal distribution, by virtue of the central limit theorem, and can be approximated by the Poisson distribution, as we have just studied. The normal approximation works well when np and n(1 - p) are large; the rule of thumb is that both should be at least 5. The Poisson approximation works well when n is large, p small so that np is of moderate size.
11. In the
binomial timeline experiment, set n = 40 and p = 0.1 and run the simulation
1000 times with an update frequency of 10. Compute and compare each of the following:
12. In the
binomial timeline experiment, set n = 100 and p = 0.1 and run the simulation
1000 times with an update frequency of 10. Compute and compare each of the following:
13. A text file
contains 1000 words. Assume that each word, independently of the others, is misspelled
with probability p.
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 A 
[0, ), let m(A)
denote the length of A and let N(A) denote the number of random
points in A. Suppose that for some r > 0, the following axioms are
satisfied:
r as m(A) 0 (the rate property). 0 as m(A)
0 (the sparseness property).The following exercises will show that these axioms define a Poisson process. First, let
Nt = N(0, t], Pn(t) = P(Nt
= n) for t 
0, n = 0, 1, 2,
...
14. Use the axioms
to show that P0 satisfies the following differential equation and
initial condition:
15. Solve the
initial value problem in Exercise 14 to show that
P0(t) = e-rt.
16. Use the axioms
to show that Pn satisfies the following differential equation and
initial condition for n = 1, 2, ...:
17. Solve the
differential equations in Exercise 16 recursively to show that
Pn(t) = e-rt (rt)n / n! for n = 1, 2, ...
From Exercise 17, it follows that Nt has the Poisson distribution with parameter rt. Now let Tk denote the k'th arrival time for k = 1, 2, .... As before, we must have
Nt k if and only
if Tk t.
18. Show that Tk
has the gamma distribution with parameters k and r.
Finally, let Xk = Tk - Tk - 1 denote the k'th interarrival time, for k = 1, 2, ...
19. Show that the
interarrival times Xk, k = 1, 2, ... are independent and each
has the exponential distribution with parameter r.