What is the Difference Between Binomial and Poisson?

The Binomial and Poisson distributions are both used in probability theory and statistics, but they differ in the nature of the experiment, number of trials, probability of success, and probability distribution. The main differences between Binomial and Poisson distributions are:

1. Nature of the experiment: Binomial distribution deals with experiments involving a fixed number of independent trials, while the Poisson distribution focuses on events occurring over a fixed interval.
2. Number of trials: The Binomial distribution has a fixed number of trials (n), whereas the Poisson distribution has an unlimited number of trials.
3. Probability of success: In the Binomial distribution, the probability of success (p) is constant, while in the Poisson distribution, the probability of success is extremely small.
4. Outcomes: The Binomial distribution has only two possible outcomes (success or failure), whereas the Poisson distribution has an unlimited number of possible outcomes.
5. Parameters: The Binomial distribution is biparametric, featuring two parameters (n and p), while the Poisson distribution is uniparametric, characterized by a single parameter (m).
6. Mean and variance: In the Binomial distribution, the mean is greater than the variance, while in the Poisson distribution, the mean is equal to the variance.

In summary, the Binomial and Poisson distributions are both essential tools in probability theory and statistics, but they are applied to different types of experiments and situations. Understanding their differences and knowing when to apply each distribution is crucial for accurate data analysis and modeling.

Comparative Table: Binomial vs Poisson

Here is a table comparing the differences between Binomial and Poisson distributions:

For example, a Binomial distribution can be used to model a coin tossing experiment, while a Poisson distribution can be used to model the number of printing mistakes per page of a large book.