Simulation of a Poisson Process on the Real Line
In this section of the lab we will
simulate a Poisson process. Ultimately, we will use such simulations to check
property (iii) of Section 4.5 on page 216 of the text.
Let 
be independent random variables
each having the exponential distribution with inverse-scale parameter 1,
and set 
for 
.
As explained in the text, we think of 
as the failure time of the
ith component once it is installed in a given system.
We assume that the first component is
installed at time zero and that a fresh component is installed as soon as the
one currently installed fails. Then
is the failure
time of the nth component that is installed.
For 
, we will let 
be the
number of components that have been installed and failed by time t.
- Suppose we are interested in following the process up to time
t=10.
We first need to decide upon a practical
upper limit N so that with high probability
is larger than 10.
Use the fact that 
has a gamma distribution to find a value of
N such that 
.
- For the above value of N, generate 100 samples of size N from
the exponential distribution. Use these samples to produce 100 simulated
Poisson processes. Make a scatterplot of
versus t
for three of these simulated processes (for 
).
- Use the above simulations to generate 100 observations of
and 
and draw a histogram for each.
Does the relative frequency of the
numbers 
support the theoretical
distribution of 
?
Now, consider the 100 observations of 
.
Once again, does
the relative frequency of the
numbers 
support the theoretical results given in the text
about the distribution of 
?