Parameter

There is little difference in meaning between parameter and argument of a function. It is usually a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters. The best way to explain this is to illustrate it with examples.

Table of contents 1 Analytic geometry 2 Mathematical analysis 3 Probability theory 4 Statistics 5 Computer Programming

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

<math>x^2+y^2=1</math>

• "parametric" form

<math>(x,y)=(\cos t,\sin t)</math>

where t is the "parameter".

Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form

<math>F(t)=\int_{x_0(t)}^{x_1(t)}f(t)\,dt</math>

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 Gaussian distribution with mean μ and variance σ²".

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.

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 mean (<math>\overline X</math>) of a sample is an estimate of the mean (μ) of the population from which it was drawn.

Computer Programming

In programming, the term parameter takes on a specialized meaning. Parameters are a way of allowing the same sequence of commands to operate on different data without re-specifying the instructions.

For example, take the following list of instructions:

1. Take an object.
2. Break it into little pieces.
3. Throw it away.

In this case, the object that the instructions are to operate on is the parameter. If we give this process a name like Destroy, then referring to Destroy followed by the desired object will perform the actions on that object.

For instance:

1. Destroy rock.
2. Destroy cake.
3. Destroy car.

Will apply the instructions above to a rock, cake, and car respectively.

This notion is useful for both being able to reason about a sequence of processes and effectively programming a computer. Much of the theory of programming languages deals with various method of passing parameters. These include: call-by-value, call-by-reference, call-by-name.