Working with random processes
In this recipe, we will examine a simple example of a random process that models the number of bus arrivals at a stop over time. This process is called a Poisson process. A Poisson process, 
, has a single parameter, 
, which is usually called the intensity or rate, and the probability that 
takes the value 
at a given time 
is given by the following formula:
This equation describes the probability that buses have arrived by time 
. Mathematically, this equation means that 
has a Poisson distribution with the parameter 
. There is, however, an easy way to construct a Poisson process by taking sums of inter-arrival times that follow an exponential distribution. For instance, let 
be the time between the (
)-st arrival and the 
-th arrival, which are exponentially distributed with parameter 
. Now, we take the following equation:
Here, the number 
is the maximum 
such that 
. This is the construction that we will work through in this recipe. We will...