Introduction
In probability theory, the Poisson process is one of the most important and commonly utilized processes. It's a popular tool for simulating random events in time or space. The Poisson Process can be used to simulate a variety of real-life scenarios. For example, the number of accidents on the road, model the number of earthquakes in a given area, etc.
Let’s learn about the Homogeneous Poisson Process in-depth.
Homogeneous Poisson Process
To understand the Poisson process, we first need to understand about counting process.
Counting Process: A counting process, N(t), is any integer-valued process with the following properties:-
1.) N(0) = 0.
2.) N(t + s) ≥ N(t), ∀s ≥ 0.
Poisson Process is derived as a counting process here. Assume we're tracking the number of times a specific event occurs over a given period of time. (In this case, we're using time as an example.) We may also think of things like space and so on).
We can classify them as Poisson Process events if they meet the following criteria.
1.) The frequency of occurrences over disjoint time intervals is independent.
2.) The probability of a single occurrence in a short time span is proportional to the interval's length.
3.) The probability of multiple occurrences during a small time interval can be ignored.
If we use the symbol X(t) to represent the number of occurrences across a time interval of length t, then.
, where 
is the rate of occurrence.
Derivation
Let’s prove our claim that if X(t) be the number of occurrence in an interval of length t, then.
, where is the rate of occurrence.
The statement will be proven using mathematical induction. We begin by expressing the above assumptions mathematically. According to criteria 3 in a small time interval h.
, where 
tends to zero as h tends to zero or.
If is the rate of occurrence, then according to criteria 2, we get,
Take a small interval (t, t+h) and an interval (0, t). P(X(t)=n) will be abbreviated as 
. As a result, the equations above can be represented as,
So we have to prove that,
We'll start by proving the result for n=0 and n=1. Then we'll verify that if the result is correct for n=m, it'll be correct for n=m+1.
Take the interval (0, t+h). Now,
(since occurrences in the intervals (0, t) and (t, t+h) are independent) or,
Taking the limit as h tends to zero, we get,
The solution of the above differential equation is,
taking the initial condition, 
we evaluate c=0. Hence,
, So our claim is valid for n=0.
Now we try to prove it for n=1.
(We use the fact that the occurrence must be in either of the interval (0, t) and (t, t+h)), or
Again taking the limit as h tends to zero,
This is a first-order linear differential equation, and the solution is,
, where c1 is a constant.
Since, 
. We get,
c1 = 0 .
Hence
So our claim is valid for n=1. We assume that our claim is right for n=m.
We will show that it is true for n=m+1. So,
(We suppose that the m+1 occurrence can occur in a variety of ways, including m+1 occurrences in (0, t) and no occurrence in (t, t+h), or m occurrences in (0, t) and 1. an occurrence in (t, t+h), or m-j occurrences in (0, t) and j+1 occurrence in (t, t+h) for j=1 to m).
So,
Since,
for j>=1.
or,
Taking the limit as h goes to zero, we have,
This is a first-order differential equation with a solution that looks like this.
If we assume that 
we get c2 = 0.
So the final result is,
Hence the result is proved.
Thus, we have derived the pmf of no. of occurrences in a Poisson Process, a Poisson Distribution with parameter .
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