A parameter is a measurement or value on which something else depends.
For example, a parametric equaliser is a tone control circuit that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency and level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. By way of contrast, a graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.
In mathematics there is little difference in meaning between parameter and argument of a function. It is usually a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters. The best way to explain this is to illustrate it with examples.
In computing the parameters passed to a function subroutine are more normally called arguments.
In logic, by some authors (eg Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover") the parameters passed to (or operated on by) an open predicate are called parameters. Parameters locally defined within the predicate are called variables. This extra effort pays off when defining substitution (without this distinction special provision has to be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables, and when defining substitution have to distinguish between free variables and bound variables.
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In analytic geometry, curves are often given as the image of some function. The argument of the function is invariably called "the parameter". A circle of radius 1 centered at the origin can be specified in more than one form:
In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form
In probability theory, one may describe the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters. For example, one talks about "a Poisson distribution with mean value λ", or "a normal distribution with mean μ and variance σ2". The latter formulation and notation leaves some ambiguity whether σ or σ2 is the second parameter; the distinction is not always relevant.
It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution.
In statistics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In classical estimation these parameters are considered "fixed but unknown", but in Bayesian estimation they are random variables with distributions of their own.
Statistics are mathematical characteristics of samples which are used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the sample mean () is an estimate of the mean parameter (μ) of the population from which the sample was drawn.
On the computer, parameters are used to differentiate behavior or pass input data to computer programs or their subprograms. See parameter for detail.Analytic geometry
Mathematical analysis
Now, if we performed the substitution x=g(y), it would be called a "change of variable".Probability theory
Statistics
Computer
