Introduction

The Poisson Distribution is a fundamental probability distribution used to model the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It’s commonly applied in scenarios where events are rare but have a consistent average rate. Understanding the Poisson Distribution helps us make predictions about the likelihood of a specific number of events occurring within a certain timeframe or region.

Properties and Characteristics

The Poisson Distribution is characterized by the following properties:

• Events are rare and random.
• The events are independent of each other.
• The average rate of occurrence is denoted by $\lambda$ (lambda).

The probability mass function (PMF) of the Poisson Distribution is given by the formula:

$$P(X=k)=\frac{e^{-\lambda} \cdot \lambda^k}{k!}$$ Where:

• $k$ is the number of events
• $e$ is the base of the natural logarithm

Example: Consider a call center that receives an average of 4 customer service calls per hour. What’s the probability of receiving exactly 2 calls in the next hour? Solution: Given that $\lambda=4$, $k=2$, using the Poisson Distribution formula, we have:

$$ P(X=2)=\frac{e^{-4} \cdot 4^2}{2 !} \approx 0.1465$$

So, there’s approximately a 14.65% chance of receiving exactly 2 calls in the next hour.

Limitations and Assumptions:

The Poisson Distribution assumes that events occur randomly and independently, and that the average rate remains constant. It may not be suitable for scenarios with events that are not truly independent or for situations where the average rate changes.

Applications

• Insurance Claims: The Poisson Distribution can model the number of insurance claims in a given period, assuming a stable average claim rate.

• Network Traffic: It’s used to estimate the number of data packets arriving at a network router in a specific time interval.

• Earthquakes: In seismology, it’s employed to predict the number of earthquakes in a region based on historical data.