distribiutieparameters

Distribuție parameters, in statistics often referred to as distribution parameters, are numerical constants that specify the form of a probability distribution. They determine properties such as central tendency, dispersion, shape, and sometimes the support of the distribution. Broadly, parameters fall into location, scale, and shape categories. Location parameters shift the distribution along the horizontal axis; scale parameters stretch or compress it; shape parameters influence skewness and tail heaviness.

Common examples illustrate how parameters control behavior. The normal distribution uses two parameters: the mean (μ), a location parameter, and the standard deviation (σ), a scale parameter. The exponential distribution is parameterized by a rate λ (or equivalently a mean of 1/λ). The gamma distribution uses shape k and scale θ, which jointly affect both form and spread. The beta distribution on the interval [0,1] uses shape parameters α and β to control its symmetry and peak. Discrete distributions also rely on parameters, for example the binomial distribution is defined by the number of trials n and the success probability p.

Parameter estimation aims to infer these constants from data, typically through methods such as maximum likelihood estimation, method of moments, or Bayesian inference. Once estimated, parameters enable prediction, interval estimation, and hypothesis testing. Some parameterizations are interchangeable (for example rate versus scale), and different parameterizations can affect numerical stability and interpretation. It is important to ensure parameters comply with their natural constraints (such as positivity for variance-related parameters) and to consider identifiability and model fit when selecting a parameterization.

In practice, the choice and estimation of distribution parameters underpin statistical modeling, inference, and decision-making across applied fields.