kb/data/en.wikipedia.org/wiki/Parameter-1.md

title	chunk	source	category	tags	date_saved	instance
Parameter	2/4	https://en.wikipedia.org/wiki/Parameter	reference	science, encyclopedia	2026-05-05T07:24:40.755713+00:00	kb-cron

[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" 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 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 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.

=== Analytic geometry ===

In analytic geometry, a curve can be described as the image of a function whose argument, typically called the parameter, lies in a real interval. For example, the unit circle can be specified in the following two ways:

implicit form, the curve is the locus of points (x, y) in the Cartesian plane that satisfy the relation

      x
      
        2
      
    
    +
    
      y
      
        2
      
    
    =
    1.
  

{\displaystyle x^{2}+y^{2}=1.}

parametric form, the curve is the image of the function

    t
    ↦
    (
    cos
    ⁡
    t
    ,
    sin
    ⁡
    t
    )
  

{\displaystyle t\mapsto (\cos t,\sin t)}

with parameter

    t
    ∈
    [
    0
    ,
    2
    π
    )
    .
  

{\displaystyle t\in [0,2\pi ).}

As a parametric equation this can be written

    (
    x
    ,
    y
    )
    =
    (
    cos
    ⁡
    t
    ,
    sin
    ⁡
    t
    )
    .
  

{\displaystyle (x,y)=(\cos t,\sin t).}

The parameter t in this equation would elsewhere in mathematics be called the independent variable.

=== Mathematical analysis === In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form

    F
    (
    t
    )
    =
    
      ∫
      
        
          x
          
            0
          
        
        (
        t
        )
      
      
        
          x
          
            1
          
        
        (
        t
        )
      
    
    f
    (
    x
    ;
    t
    )
    
    d
    x
    .
  

{\displaystyle F(t)=\int _{x_{0}(t)}^{x_{1}(t)}f(x;t)\,dx.}

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 dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration).

=== Statistics and econometrics ===