What are the differences and connections between binomial distribution, Poisson distribution and normal distribution?

The binomial, Poisson, and normal distributions are interconnected probability models that describe different stochastic mechanisms but converge under specific mathematical conditions. The binomial distribution is discrete, modeling the number of successes in a fixed number of independent Bernoulli trials, each with an identical probability of success. Its shape is determined by two parameters: the number of trials and the success probability. The Poisson distribution is also discrete, but it models the number of events occurring in a fixed interval of time or space, characterized by a known constant average rate and the property that events occur independently. Its single parameter is the average rate, which also equals its variance. In contrast, the normal distribution is a continuous model defined by its mean and variance, describing data that clusters symmetrically around a central value, and it arises naturally from the aggregation of many small, independent effects due to the Central Limit Theorem.

The primary connections between these distributions are asymptotic. A binomial distribution with a large number of trials and a small success probability can be approximated by a Poisson distribution, a relationship formalized by the law of rare events. More fundamentally, both the binomial and Poisson distributions converge to the normal distribution under certain limits. For the binomial, if the number of trials is large and the success probability is not too close to zero or one, the distribution of the count can be approximated by a normal distribution with matching mean and variance, a consequence of the De Moivre-Laplace theorem. Similarly, for the Poisson distribution, as its rate parameter becomes large, its shape becomes increasingly symmetric and bell-shaped, allowing for an effective normal approximation. These limit theorems are not just mathematical curiosities; they provide the foundational justification for using the tractable normal distribution in statistical inference for counts when the underlying parameters are sufficiently large.

The practical implications of these differences and connections are profound in statistical modeling and hypothesis testing. Choosing the correct model depends on the data-generating mechanism: the binomial is appropriate for proportions from a fixed sample size, the Poisson for rates of occurrence, and the normal for continuous measurements or averages. However, the asymptotic connections enable critical simplifications. For instance, standard confidence intervals and tests for a proportion often rely on the normal approximation to the binomial. In quality control, Poisson limits for defect counts are routinely approximated by normal control charts when the expected count is high. Misapplication, however, carries significant risk. Using a normal approximation for a binomial process with a small sample size and an extreme probability, or for a Poisson process with a low rate, can lead to invalid probabilities, inaccurate confidence intervals, and erroneous test conclusions. Therefore, understanding the specific parametric assumptions and the validity conditions for these asymptotic bridges is essential for robust analysis.

Ultimately, these three distributions form a core triumvirate in probability theory, each representing a fundamental stochastic idea—repeated trials, rare events, and additive aggregation. Their mathematical relationships demonstrate how discrete count processes can, under the right scaling, be managed with the tools of continuous distribution theory. This synergy is central to much of applied statistics, from designing industrial sampling plans using binomial-derived normal limits to modeling traffic flow or call center data where Poisson and normal models are used interchangeably at high volumes. The analyst's task is to correctly identify the operative mechanism in the data at hand and to judiciously apply the appropriate approximating connection, thereby leveraging the simplicity of the normal distribution without violating the mathematical premises that make such an approximation valid.