A
parameter is a measurement or value on which something else depends.
Example
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.
Types of parameter
Mathematical
In
mathematics the difference in meaning between a
parameter and an
argument of a function is that the parameters are the symbols that are part of the function's
definition, while arguments refer to the symbols that are supplied to the function when it is used. The value or objects assigned to the
parameters by the corresponding arguments of a function or system are not reassigned during the function's evaluation. So, parameters are effectively constants during the evaluation or processing of that function or system. The value of arguments can change outside of the function and between function usages. This distinction, the parameter's constancy, is a key part of the meaning of a parameter in any situation, often in usage beyond just mathematics.
In some informal situations people regard it as a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters.
Computer science
When the terms
formal parameter and
actual parameter are used, they generally correspond with the
definitions used in computer science. In the definition of a function such as
:''f''(''x'') =
x + 2,
x is a formal parameter. When the function is used as in
:''y'' =
f(3) + 5,
3 is the actual parameter value that is used to solve the equation. These concepts are discussed in a more precise way in
functional programming and its foundational disciplines,
lambda calculus and
combinatory logic.
In
computing the parameters passed to a function subroutine are more normally called
arguments.
Logic
In
logic, the parameters passed to (or operated on by) an
open predicate are called
parameters by some authors (e.g. Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Parameters locally defined within the predicate are called
variables. This extra distinction 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.
Analytic geometry
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:
:
:
:where
t is the "parameter".
A somewhat more detailed description can be found
here.
Mathematical analysis
In
mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form
:
In this formula,
t is the
argument of the function
F on the left-hand side, and the
parameter that the integral depends on, on the right-hand side. The quantity
x is a
dummy variable or
variable (or parameter) of integration. Now, if we performed the substitution
x=''g''(''y''), it would be called a "change of variable".
Probability theory
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 σ
2". The latter formulation and notation leaves some ambiguity whether σ or σ
2 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.
Statistics
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.
It is possible to make statistical inferences without assuming a particular
parametric family of probability distributions. In that case, one speaks of
non-parametric statistics as opposed to the
parametric statistics described in the previous paragraph.
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.
Computer
On the
computer, parameters are used to differentiate behavior or pass input data to computer programs or their subprograms. See
parameter (computer science) for detail.
See also: Parametrization (i.e.
coordinate system).
Category:Mathematical terminology
ja:媒介変数
nl:Parameter