# User Contributed Dictionary

- Plural of parameter

# Extensive Definition

In mathematics, statistics, and the
mathematical sciences, a
parameter (G:
auxiliary measure) is a quantity that defines certain
characteristics of systems or functions.
Often represented by θ in general form, other symbols carry
standard, specific meanings. When evaluating the function over a
domain or determining the
response of the system over a period of time, the independent
variables are varied, while the parameters are held constant.
The function or system may then be reevaluated or reprocessed with
different parameters, to give a function or system with different
behavior.

## Example

- In a section on frequently misused words in his book The Writer's Art, James J. Kilpatrick quoted a letter from a correspondent, giving examples to illustrate the correct use of the word parameter:

- A parametric equaliser is an audio filter 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 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. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.

- If asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different graphical appearance. The a can therefore be considered to be a parameter: less variable than the variable x, but less constant than the constant 2.

## Parameters in various contexts in math and science

### Mathematical functions

Mathematical functions typically can have one or more variables and zero or more parameters. The two are often distinguished by being grouped separately in the list of arguments that the function takes:- f(x_1, x_2, \dots; a_1, a_2, \dots) = \cdots\,

The symbols before the semicolon in the
function's definition, in this example the x's, denote variables,
while those after it, in this example the a's, denote
parameters.

Strictly speaking, parameters are denoted by the
symbols that are part of the function's definition, while arguments
are the values that are supplied to the function when it is used.
Thus, a parameter might be something like "the ratio of the
cylinder's radius to its height", while the argument would be
something like "2" or "0.1".

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.

### 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:- implicit form

- x^2+y^2=1

- parametric form

- (x,y)=(\cos t,\sin t)
- where t is the parameter.

### Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form- F(t)=\int_^f(x;t)\,dx.

### 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". The function defining the distribution (the probability mass function) is:- f(k;\lambda)=\frac.

For instance, suppose we have a radioactive sample that
emits, on average, five particles every ten minutes. We take
measurements of how many particles the sample emits over ten-minute
periods. The measurements will exhibit different values of k, and
if the sample behaves according to Poisson statistics, then each
value of k will come up in a proportion given by the probability
mass function above. From measurement to measurement, however, λ
remains constant at 5. If we do not alter the system, then the
parameter λ is unchanged from measurement to measurement; if, on
the other hand, we modulate the system by replacing the sample with
a more radioactive one, then the parameter λ would increase.

Another common distribution is the normal
distribution, which has as parameters the mean μ and the
variance σ².

It is possible to use the sequence of moments
(mean, mean square, ...) or cumulants (mean, variance, ...)
as parameters for a probability distribution.

### Statistics and econometrics

In statistics and econometrics, 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. For example,
Spearman is a non-parametric test as it is computed from the
order of the data regardless of the actual values, whereas
Pearson is a parametric test as it is computed directly from
the data and can be used to derive a mathematical
relationship.

Statistics are
mathematical characteristics of samples which can be used as
estimates of parameters, mathematical characteristics of the
populations from which the samples are drawn. For example, the
sample mean (\overline X) can be used as an estimate of the mean
parameter (μ) of the population from which the sample was
drawn.

## Other fields

Other fields use the term "parameter" as well, but with a different meaning.### 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.### Engineering

In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. For example an airliner flight data recorder may record 88 different items, each termed a parameter. This usage isn't consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel."Speaking generally, properties are those
physical quantities which directly describe the physical attributes
of the system; parameters are those combinations of the properties
which suffice to determine the response of the system. Properties
can have all sorts of dimensions, depending upon the system being
considered; parameters are dimensionless, or have the dimension of
time or its reciprocal." John D. Trimmer, 1950, Response of
Physical Systems (New York: Wiley), p. 13.

The term can also be used in engineering
contexts, however, as it is typically used in the physical
sciences.

### 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 or just the value of f(3),

3 is the actual parameter value that is
substituted for x, the formal parameter, in the function
definition. These concepts are discussed in a more precise way in
functional
programming and its foundational disciplines, lambda
calculus and combinatory
logic.

In computing, parameters are
often called arguments, and the two words are used interchangeably.
However, some computer languages such as C define argument to mean
actual parameter (i.e., the value), and parameter to mean formal
parameter.

## See also

- Parametrization (i.e., coordinate system)
- Parametrization (climate)
- Parsimony (with regards to the trade-off of many or few parameters in data fitting)

parameters in Arabic: وسيط

parameters in German: Parameter
(Mathematik)

parameters in Esperanto: Parametro

parameters in French: Paramètre

parameters in Italian: Parametro
(statistica)

parameters in Hebrew: פרמטר

parameters in Dutch: Parameter

parameters in Japanese: 媒介変数

parameters in Norwegian: Parameter

parameters in Quechua: Kuskanachina tupu

parameters in Russian: Параметр

parameters in Simple English: Parameter

parameters in Slovak: Parameter

parameters in Serbian: Параметар

parameters in Sundanese: Paraméter

parameters in Swedish: Parameter

parameters in Vietnamese: Tham
số