Parameter

A parameter is a characteristic used to define or classify a system. A system might be an object, a project, an event, or a specific situation. Parameters are the essential elements used to identify a system. They are also critical when evaluating a system's performance, status, or condition. In technical fields, the term has specific meanings. You will find it used in mathematics, computer programming, engineering, and statistics. It also appears in linguistics and electronic musical composition. In non-scientific settings, people use it to describe boundaries. For example, someone might discuss "test parameters" or "game play parameters."

In mathematical modeling, parameters are the values that describe a system. When a system is represented by equations, these values act as the parameters. In the field of mechanics, parameters include mass and dimensions. For solid bodies, the shape is also a parameter. When modeling the movement of fluids, scientists use density and viscosity as parameters. There are often many ways to choose these values. This process of selecting a set of parameters is called parametrization. For example, you can describe a position on Earth using angular coordinates. These are latitude and longitude, which describe movement along circles. You could also use directional distance from a known point. This might mean being 10km North-Northwest of Toronto. Such methods are also used in the modelization of geographic areas for map drawing.

Mathematical functions use both variables and parameters. A function has one or more arguments called variables. A function definition can also include parameters. However, parameters are not listed among the arguments the function takes. When parameters are present, they define a whole family of functions. Every valid set of parameter values creates a different function in that family. Consider a general quadratic function defined by the equation y = ax² + bx + c. Here, x is the variable that designates the function's argument. The values a, b, and c are parameters, also called coefficients. These coefficients determine which specific quadratic function is being considered. A parameter can even be part of a function's name. A base-b logarithm uses b as a parameter to indicate which function is used. In this case, b is not an argument but remains constant during certain operations.

Changing the status of a symbol between a parameter and a variable changes the mathematical object. For instance, the falling factorial power n(k) behaves differently depending on which symbol is the parameter. If k is the parameter, the expression is a polynomial function of n. If n is the parameter, it is not a polynomial function of k. In this second case, it is only defined for non-negative integer arguments. Formal mathematics often starts with a function of several variables. This is the most fundamental object. Mathematicians then define functions with fewer variables using a process called currying.

We can see the difference between variables and parameters in everyday life. Imagine you are calculating your total income. Income equals your wage multiplied by the hours you work. In this scenario, the hours worked is an independent variable because it changes easily. The wage is a parameter because it is more static. Another example involves the speed of a car. The speed is a dependent variable that changes based on the gas pedal. The pedal position is the independent variable. If an engineer changes the lever arms of the car's linkage, they have changed a parameter. The speed will still depend on the pedal, but it will do so in a different manner.

In the field of statistics, parameters help us understand entire populations. A model often consists of equations that describe physical situations. These equations contain both variables and parameters. Variables are quantities that can be measured independently in an experiment. Parameters are often "constants" that represent inherent properties of nature or equipment. In frequentist estimation, parameters are viewed as fixed but unknown. In Bayesian estimation, they are treated as random variables with their own distributions. Statisticians use a "statistic" to estimate a population parameter. For example, the sample mean is an estimator for the population mean, which is the estimand. Similarly, sample variance is used to estimate the population variance parameter.

Probability theory also relies heavily on parameters to describe patterns. A random variable might belong to a family of probability distributions. These distributions are distinguished by their specific parameters.

A Poisson distribution is defined by a parameter called lambda (λ). This parameter represents the mean number of observations for a phenomenon. For example, a radioactive sample might emit five particles every ten minutes on average. In this case, lambda is 5. When you take measurements, the number of particles observed is the variable, k. Even if k changes from one measurement to the next, the parameter lambda remains constant. This shows how a parameter characterizes the system itself without being altered by the observations.