Statistical Distributions
10 probability distributions with formulas, properties, examples, and guidance on when to use each one.
Normal Distribution
The normal distribution, also called the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric about its mean. It is the most important distribution in statistics because of the Central Limit Theorem, which states that the sum of many independent random variables tends toward a normal distribution regardless of the underlying distribution. Approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is one of the most commonly used discrete distributions in statistics and is fundamental to hypothesis testing for proportions. The distribution arises whenever you count the number of times an event occurs in a fixed number of independent attempts.
Poisson Distribution
The Poisson distribution models the number of events occurring in a fixed interval of time, space, or other continuous domain, given that events occur independently at a constant average rate. It is widely used in queueing theory, reliability engineering, and epidemiology. The distribution is uniquely characterized by its single parameter λ, which represents both the mean and the variance.
Student's t-Distribution
The Student's t-distribution is a continuous probability distribution that arises when estimating the mean of a normally distributed population using a small sample size and an unknown population standard deviation. It was developed by William Sealy Gosset under the pseudonym 'Student.' The t-distribution has heavier tails than the normal distribution, reflecting the extra uncertainty from estimating the standard deviation. As the degrees of freedom increase, it converges to the standard normal distribution.
Chi-Square Distribution
The chi-square (χ²) distribution is a continuous probability distribution that arises as the sum of squares of independent standard normal random variables. It is right-skewed and takes only non-negative values. The chi-square distribution is fundamental to statistical inference, appearing in goodness-of-fit tests, tests of independence in contingency tables, and confidence intervals for population variances. The shape is determined entirely by its degrees of freedom parameter.
F-Distribution
The F-distribution is a continuous probability distribution that arises as the ratio of two independent chi-square random variables, each divided by their respective degrees of freedom. Named after Sir Ronald Fisher, it is the cornerstone distribution for analysis of variance (ANOVA) and for comparing the variances of two populations. The F-distribution is right-skewed, takes only positive values, and is characterized by two degrees of freedom parameters.
Exponential Distribution
The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is the continuous counterpart of the geometric distribution and is widely used in reliability engineering, queueing theory, and survival analysis. The exponential distribution is the only continuous distribution with the memoryless property.
Continuous Uniform Distribution
The continuous uniform distribution assigns equal probability to all values in a specified interval [a, b]. It is the simplest continuous distribution and represents complete ignorance about which value in the interval is more likely. The uniform distribution serves as a baseline for random number generation and appears in many theoretical contexts, including as a component in simulation methods and probability integral transforms.
Geometric Distribution
The geometric distribution models the number of independent Bernoulli trials needed to achieve the first success. It is the discrete analog of the exponential distribution and the only discrete distribution with the memoryless property. The geometric distribution is commonly used in quality control to model the number of inspections until the first defective item is found, and in reliability to model the number of uses until first failure.
Hypergeometric Distribution
The hypergeometric distribution models the number of successes in a sample drawn without replacement from a finite population. Unlike the binomial distribution, where each trial is independent, the hypergeometric accounts for the changing composition of the population as items are drawn. It is used extensively in quality control, ecological capture-recapture studies, and combinatorial probability problems like card-drawing scenarios.
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