About the relationship between Bernoulli, Binomial, Gaussian distribution?

The relationship between the Bernoulli, Binomial, and Gaussian distributions is a foundational progression in probability theory, moving from a single binary trial to the aggregation of many such trials and finally to a powerful continuous approximation. A Bernoulli distribution models the outcome of a single trial with two possible results, typically success (with probability *p*) or failure (with probability *1-p*). Its simplicity is its strength, providing the basic building block. When we consider a fixed number *n* of independent and identically distributed Bernoulli trials, the total number of successes follows a Binomial distribution, characterized by parameters *n* and *p*. This shift from a single event to a sum is the first critical link, as the Binomial distribution's probability mass function is explicitly derived from the Bernoulli process, calculating the probability of achieving exactly *k* successes.

The connection to the Gaussian, or normal, distribution arises through the Central Limit Theorem (CLT), a cornerstone of statistical theory. The CLT states that the sum (or average) of a large number of independent, identically distributed random variables—each with finite mean and variance—will approximately follow a Gaussian distribution, regardless of the original variable's distribution. Since a Binomial random variable is precisely such a sum (*n* independent Bernoulli variables), its distribution converges to a Gaussian as *n* becomes large. The parameters of this approximating normal distribution are mean *μ = np* and variance *σ² = np(1-p)*. This approximation is most accurate when *p* is not too close to 0 or 1, with a common rule of thumb being that both *np* and *n(1-p)* exceed 5 or 10.

This relationship is not merely theoretical but has profound practical implications for inference and computation. Calculating exact Binomial probabilities for large *n* can be computationally intensive, while the Gaussian approximation allows for the use of well-tabulated standard normal probabilities and simplifies hypothesis testing and confidence interval construction for proportions. The mechanism underpinning this is the De Moivre-Laplace theorem, a specific historical precursor to the general CLT, which formally established the normal approximation to the binomial. It is crucial to recognize the boundaries of this approximation; for extreme *p* or small *n*, the symmetric, continuous normal curve may poorly fit the discrete, potentially skewed Binomial distribution, in which case other approximations like the Poisson may be more appropriate.

Thus, these three distributions form a conceptual and practical hierarchy: the Bernoulli provides the elemental random event, the Binomial describes the exact distribution of counts from repeated events, and the Gaussian offers a versatile continuous model for that count when the number of trials is sufficiently large. This progression elegantly demonstrates how complex aggregate behavior emerges from simple components and how limiting theorems enable powerful simplifications for statistical analysis, fundamentally linking discrete probability with continuous distribution theory.