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In some non-technical contexts or in . In ''auxiliary measure'') are quantities that define certain relatively constant characteristics of systems or Function s. 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 Variable s 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. Loosely speaking, the term parameter is used for an argument which is intermediate in status between a Variable and a Constant . EXAMPLE
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: : The symbols before the semicolon in the function's definition, in this example the 's, denote variables, while those after it, in this example the '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 , Curve s 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:
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: :where ''t'' is the ''parameter''. A somewhat more detailed description can be found at Parametric Equation . Mathematical analysis In Mathematical Analysis , one often considers "integrals dependent on a parameter". These are of the form : In this formula, ''t'' is on the left-hand side the argument of the function ''F'', and it is on the right-hand side the ''parameter'' that the integral depends on. When evaluating the integral, ''t'' is held constant, and so it considered a parameter. If we are interested in the value of ''F'' for different values of ''t'', then, we now consider it to be a variable. The quantity ''x'' is a ''dummy variable'' or ''variable of integration'' (confusingly, also sometimes called a ''parameter of integration''). Probability theory In Probability Theory , one may describe the 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: : 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 ''k1'' occurrences, we plug it into the function to get . Without altering the system, we can take multiple samples, which will have a range of values of ''k'', but the system will always be characterized by the same λ. 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 Cumulant s (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. Statistic s 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'' () 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 See Also: Parameter (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'' is a '''formal parameter'''. When the function is used as in y 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 values passed to a function subroutine are more normally called ''arguments''. SEE ALSO
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