Characteristic function

One of the many important components in calculation and application of probability tools, Characteristic function is a term used to denote a wide range of concepts and principles in mathematics. It is also known as an indicator function; that is defined on a defined set of X, indicating or denoting a relationship between a single member in a subset of previously mentioned set of X. Further, all elements or members of the subset would be equivalent to a single value of 1; whereas all elements of the larger set of X that are not a part of the subset would be zero in value.

This essentially defines role as well as concept of an indicator-function or a Characteristic function. Like all other areas there are some core terms that should be necessarily understood in order to be able to execute and apply this function properly. These include a dummy variable, which is a term exclusive to statistics; proper subset, which would be a collective term for all properties that are essential to be present for any set to be a perfect subset of another. Above application of a characteristic function was only with respect to set theory in mathematics.

When, however, subject of Characteristic function and its use strictly in probability is concerned, it has a different meaning. Thus, a characteristic function on a real-line, (that is a number line, with real numbers on it); is such that it involves several terms and concepts: random variable and expected value. There besides these, there are some other areas where it is also used: such as characteristic function as used in convex analysis, characteristic function in statistical mechanics, where it undergoes a slight alteration and becomes a characteristic state function; characteristic polynomial whereby it would be used in the linear branch of algebra. It is also used in game theory, under co-operative game-theory.

However, if you were to concentrate on its use solely in probability, there are some interesting facts that are brought to the fore. It is used as a moment-generating-function as well, in probability. These are Fourier form of a random variable’s distribution in probability is given name of a characteristic function; each function has the power to uniquely define and determine distribution in probability; that function of sum of all random variables that are independent, is actually equivalent to product of all individual functions of each variable (random).

Moreover, convergence in probability-distribution is actually equal to convergence of each individual point which corresponds to characteristic functions. Experts and mathematicians have found that it is far more convenient working using sums or additions, with aid of natural logarithms, instead of going in for the other option of products and multiplication of each individual element. There are specific formulae that you would have to apply when you are going in for the former use of kind of application or methodology of natural logarithm, as opposed to that of adding up each individual element. The area and scope for calculation of a characteristic function is vast; and encompasses innumerable principles and terms that are all interconnected, such that each affects determination of the other.