Several named probability distributions serve as building blocks for statistical models. The Bernoulli distribution models a single trial with two outcomes, success with probability \(p\) and failure with probability \(1 - p\). Repeating such trials independently yields the binomial distribution, which counts the number of successes in \(n\) Bernoulli trials; it has mean \(np\) and variance \(np(1-p)\). The Poisson distribution models the count of events occurring in a fixed interval at a constant rate \(\lambda\), with both its mean and variance equal to \(\lambda\). The geometric distribution counts the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials.
For continuous variables, the normal distribution is the most important: a symmetric, bell-shaped curve fully described by its mean \(\mu\) and standard deviation \(\sigma\). The standard normal is the special case with mean 0 and standard deviation 1, and the 68-95-99.7 rule states that about 68%, 95%, and 99.7% of observations fall within 1, 2, and 3 standard deviations of the mean, respectively. The t-distribution resembles the normal but has heavier tails, making it appropriate when the sample size is small or the population standard deviation is unknown. The chi-square distribution is the distribution of a sum of squared standard normal variables and is used in tests involving variance or categorical data, while the F-distribution, the ratio of two scaled chi-square variables, underpins ANOVA and many regression diagnostics. Two theoretical results unify much of this material. The central limit theorem states that the sampling distribution of the mean approaches a normal distribution as sample size grows, regardless of the population's underlying shape; this is what justifies normal-based inference for many real-world data. The law of large numbers guarantees that the sample mean converges to the population mean as the sample grows, so averages become more reliable with more data.