Poisson Distribution

While the Binomial distribution counts successes in a fixed number of trials, the Poisson Distribution counts the number of times an event occurs in a fixed interval of time or space.

1. What defines a Poisson Process?

For a variable to follow a Poisson distribution, it must meet three specific criteria:

2. The Mathematical Formula

The Probability Mass Function (PMF) of a Poisson distribution tells us the probability of observing $k$ events in an interval, given the average rate $\lambda$ (Lambda).

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

Key Parameters:

• $\lambda$ (Lambda): The average number of events per interval.
• $k$: The actual number of occurrences we want to find the probability for ($0, 1, 2, \dots$).
• $e$: Euler's constant ($\approx 2.718$).

Properties:

• Mean ($\mu$): $\lambda$
• Variance ($\sigma^2$): $\lambda$

Unique Property

The Poisson distribution is unique because its Mean and Variance are equal. If your data's variance is much higher than its mean (Overdispersion), a simple Poisson model might not be enough!

3. Poisson as the "Limit" of Binomial

The Poisson distribution is actually a special case of the Binomial distribution. When you have a massive number of trials ($n \to \infty$) and a very small probability of success ($p \to 0$), the Binomial distribution $B(n, p)$ turns into a Poisson distribution $P(\lambda)$ where $\lambda = np$.

4. Why this matters in Machine Learning

A. Modeling Rare Events

Poisson is used to model things like the number of credit card frauds per day or the number of times a server crashes in a month. These are "rare" relative to the total number of opportunities for them to happen.

B. Natural Language Processing (NLP)

In some classical NLP models, the frequency of a rare keyword in a document is modeled using a Poisson distribution. This helps in identifying if a word appears more often than "random chance" would suggest.

C. Traffic Prediction

Predicting the number of queries reaching a database or the number of users logging into an app in a specific minute. This is vital for Auto-scaling infrastructure in cloud computing.

D. Poisson Regression

This is a type of Generalized Linear Model (GLM) used when the target variable ($y$) is a count (e.g., predicting the number of insurance claims or the number of items sold).

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5. Summary Comparison

Feature	Binomial	Poisson
Interval	Fixed number of trials ($n$)	Fixed unit of time/space
Outcomes	Binary (Success/Failure)	Non-negative counts ($0, 1, \dots$)
Key Parameter	$p$ (Probability)	$\lambda$ (Average rate)
Limit	$n$ is finite	$n$ is infinite (theoretically)

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Now that we've covered the most important discrete and continuous distributions, how do we use them to actually evaluate a model's performance? We need to look at how we measure the distance between two distributions.