Editing Parameter (section)

==Mathematical functions{{anchor|Mathematics}}==
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 [[mathematics]], [[statistics]], and the mathematical [[science]]s, a ''parameter'' is a quantity that serves to relate [[function (mathematics)|function]]s and [[variable (mathematics)|variable]]s using a common variable when such a relationship would be difficult to explicate with an [[equation]].
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[[Mathematical function]]s have one or more [[Argument of a function|arguments]] that are designated in the definition by [[variable (mathematics)|variable]]s. A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a general [[quadratic function]] by declaring
:<math>f(x)=ax^2+bx+c</math>;

Here, the variable ''x'' designates the function's argument, but ''a'', ''b'', and ''c'' are parameters (in this instance, also called ''[[coefficient]]s'') that determine which particular quadratic function is being considered. A parameter could be incorporated into the function name to indicate its dependence on the parameter. For instance, one may define the base-''b'' logarithm by the formula
:<math>\log_b(x)=\frac{\log(x)}{\log(b)}</math>
where ''b'' is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, be a constant when considering the [[derivative (mathematics)|derivative]] <math>\textstyle\log_b'(x) = (x\ln(b))^{-1}</math>.

In some informal situations it is a matter of convention (or historical accident) whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object. For instance, the notation for the [[falling factorial power]]
:<math>n^{\underline k}=n(n-1)(n-2)\cdots(n-k+1)</math>,
defines a [[Polynomial#Polynomial functions|polynomial function]] of ''n'' (when ''k'' is considered a parameter), but is not a polynomial function of ''k'' (when ''n'' is considered a parameter). Indeed, in the latter case, it is only defined for non-negative integer arguments. More formal presentations of such situations typically start out with a function of several variables (including all those that might sometimes be called "parameters") such as
:<math>(n,k) \mapsto n^{\underline{k}}</math>
as the most fundamental object being considered, then defining functions with fewer variables from the main one by means of [[currying]].

Sometimes it is useful to consider all functions with certain parameters as ''parametric family'', i.e. as an [[indexed family]] of functions. Examples from probability theory [[#Probability theory|are given further below]].

===Examples===
* 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'':

<blockquote>W.M. Woods ... a mathematician ... writes ... "... a variable is one of the many things a ''parameter'' is not." ... The dependent variable, the speed of the car, depends on the independent variable, the position of the gas pedal.
</blockquote>
<blockquote>[Kilpatrick quoting Woods] "Now ... the engineers ... change the lever arms of the linkage ... the speed of the car ... will still depend on the pedal position ...'' but in a ... different manner''.  You have changed a parameter"</blockquote>

* 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''&nbsp;=&nbsp;''ax''<sup>2</sup>, 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 relation between ''x'' and ''y''. Thus ''a'' is a parameter: it is less variable than the variable ''x'' or ''y'', but it is not an explicit constant like the exponent&nbsp;2. More precisely, changing the parameter ''a'' gives a different (though related) problem, whereas the variations of the variables ''x'' and ''y'' (and their interrelation) are part of the problem itself.
* In calculating income based on wage and hours worked (income equals wage multiplied by hours worked), it is typically assumed that the number of hours worked is easily changed, but the wage is more static. This makes ''wage'' a parameter, ''hours worked'' an [[independent variable]], and ''income'' a [[dependent variable]].

===Mathematical models===
In the context of a [[mathematical model]], such as a [[probability distribution]], the distinction between variables and parameters was described by Bard as follows:
:We refer to the relations which supposedly describe a certain physical situation, as a ''model''. Typically, a model consists of one or more equations. The quantities appearing in the equations we classify into ''variables'' and ''parameters''. The distinction between these is not always clear cut, and it frequently depends on the context in which the variables appear. Usually a model is designed to explain the relationships that exist among quantities which can be measured independently in an experiment; these are the variables of the model. To formulate these relationships, however, one frequently introduces "constants" which stand for inherent properties of nature (or of the materials and equipment used in a given experiment). These are the parameters.<ref>{{cite book |first=Yonathan |last=Bard |year=1974 |title=Nonlinear Parameter Estimation |page=11 |location=New York |publisher=[[Academic Press]] |isbn=0-12-078250-2 }}</ref>

===Analytic geometry===
{{Main|Parametric equation|Parametrization (geometry)}}

In [[analytic geometry]], a [[curve]] can be described as the image of a function whose argument, typically called the ''parameter'', lies in a [[interval (mathematics)|real interval]].

For example, the [[unit circle]] can be specified in the following two ways:

* ''implicit'' form, the curve is the locus of points {{math|(''x'', ''y'')}} in the [[Cartesian coordinate system|Cartesian plane]] that satisfy the relation <math display=block>
x^2 + y^2 = 1.
</math>
* ''parametric'' form, the curve is the image of the function <math display=block>
t \mapsto (\cos t, \sin t)
</math><p>with parameter <math>t \in [0, 2\pi).</math> As a [[parametric equation]] this can be written</p><math display=block>
(x,y)=(\cos t,\sin t).
</math><p>The parameter {{mvar|t}} in this equation would elsewhere in mathematics be called the ''[[independent variable]]''.</p>

===Mathematical analysis===
In [[mathematical analysis]], integrals dependent on a parameter are often considered. These are of the form
:<math>F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx.</math>
In this formula, ''t'' is the argument of the function ''F'', and on the right-hand side the ''parameter'' on which the integral depends. When evaluating the integral, ''t'' is held constant, and so it is considered to be a parameter.  If we are interested in the value of ''F'' for different values of ''t'', we then consider ''t'' to be a variable. The quantity ''x'' is a ''[[Bound variable|dummy variable]]'' or ''variable of integration'' (confusingly, also sometimes called a ''parameter of integration'').

===Statistics and econometrics===
{{main|Statistical parameter}}
In [[statistics]] and [[econometrics]], the probability framework above still holds, but attention shifts to [[statistical estimation|estimating]] the parameters of a distribution based on observed data, or [[Hypothesis testing|testing hypotheses]] about them. In [[Frequentist inference|frequentist estimation]] parameters are considered "fixed but unknown", whereas in [[Bayesian probability|Bayesian estimation]] they are treated as random variables, and their uncertainty is described as a distribution.{{Citation needed|date=July 2009}}<ref>{{Cite web |last=Efron |first=Bradley |date=2014-09-10 |title=Frequentist Accuracy of Bayesian Estimates |url=https://www.researchgate.net/publication/265339596 |access-date=2023-04-12 |website=researchgate.net}}</ref>

In [[estimation theory]] of statistics, "statistic" or [[estimator]] refers to samples, whereas "parameter" or [[estimand]] refers to populations, where the samples are taken from. A [[statistic]] is a numerical characteristic of a sample that can be used as an estimate of the corresponding parameter, the numerical characteristic of the [[statistical population|population]] from which the sample was drawn.

For example, the [[sample mean]] (estimator), denoted <math>\overline X</math>, can be used as an estimate of the ''mean'' parameter (estimand), denoted ''μ'', of the population from which the sample was drawn. Similarly, the [[sample variance]] (estimator), denoted ''S''<sup>2</sup>, can be used to estimate the ''variance'' parameter (estimand), denoted ''σ''<sup>2</sup>, of the population from which the sample was drawn. (Note that the sample standard deviation (''S'') is not an unbiased estimate of the population standard deviation (''σ''): see [[Unbiased estimation of standard deviation]].)

It is possible to make statistical inferences without assuming a particular parametric family of [[probability distribution]]s. In that case, one speaks of ''[[non-parametric statistics]]'' as opposed to the [[parametric statistics]] just described. For example, a test based on [[Spearman's rank correlation coefficient]] would be called non-parametric since the statistic is computed from the rank-order of the data disregarding their actual values (and thus regardless of the distribution they were sampled from), whereas those based on the [[Pearson product-moment correlation coefficient]] are parametric tests since it is computed directly from the data values and thus estimates the parameter known as the [[Correlation and dependence|population correlation]].

===Probability theory===
[[File:Poisson pmf.svg|thumb|right|These traces all represent Poisson distributions, but with different values for the parameter &lambda;]]In [[probability theory]], one may describe the [[probability distribution|distribution]] of a [[random variable]] as belonging to a ''family'' of [[probability distribution]]s, 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:
:<math>f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!}.</math>
This example nicely illustrates the distinction between constants, parameters, and variables. ''e'' is [[Euler's number]], a fundamental [[mathematical constant]]. The parameter λ is the [[mean]] number of observations of some phenomenon in question, a property characteristic of the system.  ''k'' is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample.  If we want to know the probability of observing ''k''<sub>1</sub> occurrences, we plug it into the function to get <math>f(k_1 ; \lambda)</math>.  Without altering the system, we can take multiple samples, which will have a range of values of ''k'', but the system is always characterized by the same λ.

For instance, suppose we have a [[radioactivity|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 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.

In these above examples, the distributions of the random variables are completely specified by the type of distribution, i.e. Poisson or normal, and the parameter values, i.e. mean and variance.  In such a case, we have a parameterized distribution.

It is possible to use the sequence of [[moment (mathematics)|moments]] (mean, mean square, ...) or [[cumulant]]s (mean, variance, ...) as parameters for a probability distribution: see [[Statistical parameter]].