The Poisson Distribution

Modeling Arrival Rates

The Poisson Distribution is used for events that happen randomly in time or space, but at a constant average rate ( $\lambda$ ). It is the discrete counterpart to the continuous Exponential distribution.

✦Intuition

The Distribution of 'Rare' Events

Poisson is perfect for things that happen independently and where the chance of two events happening at the exact same millisecond is effectively zero:

The number of meteors seen in an hour ( $\lambda = 5$ ).
The number of typos on a page of a book ( $\lambda = 0.5$ ).
The number of customer arrivals in a night ( $\lambda = 100$ ).

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Parameters

$\lambda$ (Lambda): The average number of events in the given time or space interval.
$k$ : The specific number of events we want to find the probability for ( $k \in \{0, 1, 2, \dots\}$ ).

Core Properties

The Poisson distribution has a very unique and elegant property: its mean and variance are identical.

Mean ( $E[X]$ ): $\lambda$
Variance ( $Var(X)$ ): $\lambda$

Poisson Distribution

λ (lambda) is the 'average' rate. If λ=5, you'll most often see 4 or 5 events, but the distribution allows for the rare chance of seeing 10+.

Avg Rate λ = 2Avg Rate λ = 5

Relationship to Binomial

The Poisson distribution can be derived as a limit of the Binomial distribution. If you have a Binomial experiment where the number of trials $n$ is very large ( $n \to \infty$ ) and the probability of success $p$ is very small ( $p \to 0$ ), but the expected number of successes $np$ remains constant (let's call it $\lambda$ ), the Binomial distribution converges to the Poisson distribution. This is often called the Law of Rare Events.

✦Intuition

Continuous Space, Discrete Events

Why does the Poisson distribution lack a distinct 'n' trials parameter? Because it models events happening in continuous time or space. You can divide an hour into infinitely many microscopic milliseconds. In each millisecond, the chance of an event happening is near zero, but across the whole hour, they add up to an expected rate $\lambda$ . This is exactly why it represents the infinite limit of the Binomial distribution!

Advanced Practice

Example 1: Server Requests

medium

A web server receives an average of 4 requests per minute ( $\lambda = 4$ ). What is the probability that it receives exactly 6 requests in a given minute?

Example 2: No Errors Allowed

hard

A fiber optic cable experiences an average of 1.5 transmission errors per kilometer ( $\lambda = 1.5$ ). What is the probability that a 2-kilometer segment of this cable transmits data completely without error?

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The Multinomial Distribution

Course Progression

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Continuous Distributions