Parameter (mathematics)
In mathematics, a parameter is a measurable characteristic or feature that plays a crucial role in various mathematical contexts, particularly in functions. The term originates from the Greek word "parametron," meaning "subsidiary measure." In the context of a mathematical function, parameters are typically fixed quantities that help define the relationship between inputs and outputs. For example, in the quadratic function \( f(x) = ax^2 + bx + c \), the values \( a \), \( b \), and \( c \) are parameters, while \( x \) is the variable that changes.
Parameters also help categorize functions into groups known as parametric families, such as the normal distribution in probability theory, which is defined by parameters like mean (\( \mu \)) and variance (\( \sigma^2 \)). Additionally, parameters appear in parametric equations, where the coordinates of points on a curve are expressed in terms of a variable parameter. Parameters are widely used in statistics, particularly in parametric estimation, which involves approximating parameter values from data samples, as seen in polling and clinical trials. Furthermore, the concept includes nuisance parameters, which are not the primary focus of analysis but must be considered in conjunction with the parameters of interest.
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Parameter (mathematics)
The word "parameter" is derived from the Greek parametron, meaning "subsidiary measure." In mathematics and the sciences, parameter refers to measurable elements, characteristics, or features of a system. This concept comes into play in many different contexts in mathematics, the most familiar being mathematical functions.
A function is a relation between a set of inputs and a set of outputs, with each input related to one and only one output, expressed, for instance, as f(x) ("f of x"). The input, x, is a variable, an element expected to change, while other elements in the function are assumed to be fixed and are considered parameters. In the quadratic functionf(x) = ax2 + bx + c, for instance, x is the only element designated as a variable, while a, b, and c are considered parameters. The concept of the function, its input, and its parameters originated in the late seventeenth century and was developed by Leonhard Euler, Jean-Baptiste Joseph Fourier, George Boole, and others in the eighteenth and nineteenth centuries.
Overview
Functions that use the same set of parameters are sometimes grouped together as a parametric family. In probability theory, for instance, the distribution of a random variable may be considered a member of a family of probability distributions that share certain parameters. One such family is the parametric family called the normal distribution, the parameters of which are the mean μ and the variance σ².
There is a separate use of the word "parameter" in mathematics: in the case of parametric equations. A parametric equation is the equation of a curve that describes the coordinates of points on the curve as functions of a variable designated as the parameter. Parametric equations of noncurve objects exist as well, with as many parameters as the dimensions in which the object exists; this includes parametric equations of parabolas, circles, and ellipses, as well as three-dimensional helices and the parametric surface of tori.
Parameters are used throughout statistics, as for instance in parametric estimation, in which the values of parameters based on data with a random component must be approximated using measurements or samples. This is how polling works, for instance: rather than ask every eligible voter who they intend to vote for and whether they intend to vote at all, a representative sample is asked, and voting results are estimated accordingly. While this strikes some as an unscientific approach, the same mathematics of parametric estimation is used in the way radar estimates the range of detected objects, and in selecting a representative patient population to conduct clinical trials of new medications. Statistics also uses the concept of the nuisance parameter: a parameter that is not the one whose value is sought or of interest, but which must be included in the same analysis as the desired parameter.
Bibliography
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