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).

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.

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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