Introduction
The characteristic function provides an alternative way for describing a random variable. Similarly to the cumulative distribution function
where 1{X ≤ x} is the indicator function — it is equal to 1 when X ≤ x, and zero otherwise. Which completely determines behavior and properties of the probability distribution of the random variable X, the characteristic function
also completely determines behavior and properties of the probability distribution of the random variable X. The two approaches are equivalent in the sense that knowing one of the functions it is always possible to find the other, yet they both provide different insight for understanding the features of the random variable. However, in particular cases, there can be differences in whether these functions can be represented as expressions involving simple standard functions.
If a random variable admits a density function, then the characteristic function is its dual, in the sense that each of them is a Fourier transform of the other. If a random variable has a moment-generating function, then the domain of the characteristic function can be extended to the complex plane, and
Note however that the characteristic function of a distribution always exists, even when the probability density function or moment-generating function do not.
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.
Read more about this topic: Characteristic Function (probability Theory)
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