Parametric family

In probability and its applications

Main article: Statistical model

For example, the probability density function fX of a random variable X may depend on a parameter θ. In that case, the function may be denoted f_X( \cdot , ; \theta) to indicate the dependence on the parameter θ. θ is not a formal argument of the function as it is considered to be fixed. However, each different value of the parameter gives a different probability density function. Then the parametric family of densities is the set of functions { f_X( \cdot , ; \theta) \mid \theta \in \Theta } , where Θ denotes the parameter space, the set of all possible values that the parameter θ can take. As an example, the normal distribution is a family of similarly-shaped distributions parametrized by their mean and their variance.

In decision theory, two-moment decision models can be applied when the decision-maker is faced with random variables drawn from a location-scale family of probability distributions.

In algebra and its applications

In economics, the Cobb–Douglas production function is a family of production functions parametrized by the elasticities of output with respect to the various factors of production.

In algebra, the quadratic equation, for example, is actually a family of equations parametrized by the coefficients of the variable and of its square and by the constant term.

References

References

1. (2006). "All of Nonparametric Statistics". Springer Texts in Statistics.
2. Mukhopadhyay, Nitis. (2000). "Probability and Statistical Inference". [[Marcel Dekker.
3. "Parameter of a distribution".