# Parameter

A parameter is a measurement or value on which something else depends.

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## 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:

• "implicit" form
[itex]x^2+y^2=1[itex]
• "parametric" form
[itex](x,y)=(\cos t,\sin t)[itex]
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

[itex]F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx[itex]

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 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.

## 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 ([itex]\overline X[itex]) 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.

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