Understanding and Applying the Poisson Process in Insurance

What is a Poisson Process?

At its heart, a Poisson process is a model for counting events that happen randomly in time but at a constant average rate. Think of claims arriving at an insurance company or customers calling a help desk. The exact timing of each event is unpredictable, but we can describe the overall process statistically.

To build our intuition, let's start with a conceptual overview.

These assumptions lead to a formal mathematical definition. A Poisson process is a type of counting process. We denote the number of events that have occurred by time $t$ as $N(t)$.

For a more rigorous definition, which will resonate with your statistical background, let's turn to a formal study note.

To summarize these key points, a process $N(t)$ is a homogeneous Poisson process with rate $\lambda$ if:

1. $N(0)=0$.
2. The process has independent increments: the number of events in non-overlapping time intervals are independent random variables.
3. The process has stationary increments: the number of events in any interval of length $t$ follows a Poisson distribution with mean $\lambda t$.

This last point is the computational core of the process. The probability of observing exactly $k$ events by time $t$ is given by the Poisson probability mass function (PMF):

A crucial property of the Poisson distribution is that its mean and variance are equal. Therefore:

• Expected number of events by time t: $E[N(t)]=\lambda t$
• Variance of the number of events by time t: $\text{Var}[N(t)]=\lambda t$

The parameter $\lambda$ is the rate of the process, with units of events per unit of time (e.g., claims per year). The parameter of the Poisson distribution itself is $\lambda t$, which represents the expected number of events in the interval of length $t$.

This graph shows three different Poisson probability distributions. The red, green, and blue dots represent the probabilities P(k) for Poisson distributions with means (and variances) of approximately 12, 26, and 40, respectively. This illustrates how the distribution spreads out and shifts to the right as the expected number of events, $\lambda t$, increases.

Inter-arrival Times and the Exponential Distribution

We have just described the Poisson process from the perspective of counting events. An equally important perspective is the time between these events. These are known as inter-arrival times or inter-event times.

For a homogeneous Poisson process with rate $\lambda$, the inter-arrival times are independent and identically distributed random variables following an Exponential distribution with mean $1/\lambda$.

As the video explains, if the number of events is Poisson, the time between them is Exponential.

• The probability density function (PDF) is $f(x)=\lambda e^{-\lambda x}$ for $x>0$.
• The mean inter-arrival time is $E[X]=1/\lambda$. This is intuitive: if claims arrive at a rate of 10 per year, the average time between claims is 1/10th of a year.
• The variance is $\text{Var}[X]=1/{\lambda }^2$.

This connection is formally stated in the actuarial study note as well.

The Memoryless Property

The Exponential distribution has a unique and critical feature called the memoryless property. This property states that the probability of waiting an additional amount of time for an event to occur is independent of how long you have already been waiting. Formally, for a random waiting time $X$:

This concept can be counter-intuitive. If you're waiting for a bus that arrives according to a Poisson process, knowing you've already waited 10 minutes gives you no information about how much longer you have to wait. The process essentially "forgets" the past.

The Poisson/Exponential Duality in Action

The link between counting events in an interval (Poisson) and the time between them (Exponential) is not just a theoretical curiosity; it's a powerful problem-solving tool. The event "the waiting time for the next claim is greater than $t$" is exactly the same as the event "zero claims have arrived in the interval $[0,t]$". We can calculate the probability of this single event in two different ways, and the results must be identical.

The following video demonstrates this equivalence beautifully with a practical example.

Let's re-state this important result. Suppose claims arrive according to a Poisson process with rate $\lambda$. What is the probability that no claims occur in the next year?

1. Poisson Perspective: The number of claims in one year, $N(1)$, is Poisson-distributed with mean $\lambda \times 1=\lambda$.
   $P(N(1)=0)=\frac{e^{-\lambda }(\lambda {)}^0}{0!}=e^{-\lambda }$
2. Exponential Perspective: The time until the first claim, $T_1$, is exponentially distributed with mean $1/\lambda$.
   $P(T_1>1)={\int }_1^{\infty }\lambda e^{-\lambda x}dx=[-e^{-\lambda x}{]}_1^{\infty }=0-(-e^{-\lambda })=e^{-\lambda }$
   The results are identical, confirming the deep connection between these two views of the same process.

Building Models with Poisson Processes

Two essential properties allow us to construct more realistic models from simple Poisson processes: summing and thinning.

1. Summing (or Superposition)

If you have multiple independent Poisson processes, the process formed by combining all their events is also a Poisson process. Its rate is simply the sum of the individual rates.

Fact: If $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates ${\lambda }_1$ and ${\lambda }_2$, then $N(t)=N_1(t)+N_2(t)$ is a Poisson process with rate $\lambda ={\lambda }_1+{\lambda }_2$.

• Application: An insurer models claims from private car policies as a Poisson process with rate ${\lambda }_{car}=50$ claims/day and claims from commercial vehicle policies with rate ${\lambda }_{comm}=20$ claims/day. The total stream of claims arriving at the company is a Poisson process with rate ${\lambda }_{total}=50+20=70$ claims/day.
  You can find the formal statement of this in the "Poisson processes" study note under "Fact 1.15 (sum of Poisson processes)".

2. Thinning (or Splitting)

If you take a Poisson process and classify each event into one of several distinct types, then the process for each type is also an independent Poisson process. The rate for each type is the original rate multiplied by the probability of that type.

Fact: If $N(t)$ is a Poisson process with rate $\lambda$, and each event is independently classified as Type A with probability $p$ or Type B with probability $1-p$, then the process of Type A events, $N_A(t)$, is a Poisson process with rate ${\lambda }_A=\lambda p$.

• Application: A health insurer receives claims at a Poisson rate of $\lambda =100$ per hour. Historically, 5% of claims are identified as fraudulent. The arrival of fraudulent claims can be modeled as a separate Poisson process with rate ${\lambda }_{fraud}=100\times 0.05=5$ fraudulent claims/hour.
  The formal statement is in the study note under "Fact 1.13 (special types in Poisson processes, or thinning)".

Conclusion

In this lesson, we have formally defined the homogeneous Poisson process and explored its core characteristics. It serves as the essential building block for modeling the timing of random discrete events in a continuous timeframe.

Key Takeaways:

• A Poisson process $N(t)$ is a counting process with independent and stationary increments.
• The number of events in an interval of length $t$ follows a Poisson distribution with mean and variance equal to $\lambda t$.
• The time between consecutive events follows an Exponential distribution with mean $1/\lambda$ and variance $1/{\lambda }^2$.
• The process exhibits the memoryless property, meaning the past waiting time does not influence future waiting time.
• Poisson processes can be combined (summing) or split (thinning), and the resulting processes are also Poisson.

We have now successfully modeled the frequency of claim arrivals. However, an insurer is concerned not just with how many claims arrive, but also with their financial impact. In our next lesson, we will assign a random claim amount, or severity, to each event in the Poisson process. This will lead us to the Compound Poisson process, the standard model for calculating the distribution of total aggregate claims.