Poisson Distribution

It is used when events happen independently of each other and at a constant average rate.

Core Concepts

If a random variable $X$ follows a Poisson distribution, we write:

Where:

• $\lambda$ (lambda) is the average number of events in the given interval (the "rate" parameter).

The Poisson Limit Theorem (The "Why")

Imagine you divide an hour into $n$ tiny sub-intervals. If $n$ is very large (say, $n = 10,000$), the probability $p$ of a call arriving in any specific millisecond is nearly zero.

As $n \to \infty$ and $p \to 0$ such that the average rate $np = \lambda$ remains constant, the Binomial PMF mathematically transforms into the Poisson PMF:

This is why we call it the "Law of Rare Events." It models scenarios where you have many opportunities for something to happen, but the individual probability of each is tiny.

Probability Mass Function (PMF)

The probability of observing exactly $k$ events in the interval is given by:

Where:

• $e$ is Euler's number ($e \approx 2.71828$).
• $k!$ is the factorial of $k$.

Assumptions of a Poisson Process

For a real-world process to be truly modeled by Poisson, it must meet these criteria:

1. $k$ can be any positive integer ($0, 1, 2, 3, \dots$) theoretically up to infinity.
2. The occurrence of one event does not affect the probability of a second event (Independence).
3. The average rate ($\lambda$) is constant over the interval.
4. Two events cannot occur at the exact same instant in time.

Key Metrics

• Expected Value (Mean): $E[X] = \lambda$
• Variance: $\text{Var}(X) = \lambda$

Real-World Examples

1. Network Traffic: The number of requests arriving at a web server per minute.
2. Customer Service: The number of calls received by a call center per hour.
3. Astronomy: The number of meteorites hitting a specific region of the atmosphere in a year.

Related Distributions

• Binomial Distribution: The "parent" of the Poisson for finite $n$.
• Exponential Distribution: If events occur according to a Poisson process with rate $\lambda$, the waiting time between events follows an Exponential distribution.
• Normal Distribution: When $\lambda$ is large (usually $> 20$), the Poisson can be approximated by a Normal distribution.

Interactive Visualization

Use the slider below to adjust the average rate ($\lambda$). Notice how the distribution is heavily right-skewed for very low values of $\lambda$ (like 1 or 2). However, as $\lambda$ grows larger, the distribution spreads out and becomes increasingly symmetrical, once again approximating the shape of a Normal distribution.

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