# Characteristic Function

Inside mathematical statistics, characteristic function of any kind of probability distribution around the real line is due to the following formulation, where X will be any random variable with all the distribution.

$\phi_x $(t) = E(e$^itX $)

Characteristic functions can be used as part connected with procedures for size probability distributions to types of data. Estimation procedures can be purchased which match this theoretical characteristic function on the empirical characteristic operate, calculated from your data. If a random variable admits the latest probability density function, then the characteristic function could be the inverse Fourier transform within the probability density function. As a result it affords the cornerstone of a different path to analytical results which have a practical working directly getting probability density operates or cumulative submitting functions.

There are specially simple results to the characteristic features regarding distributions defined from your weighted sums about random variables.

You'll find relations between the behavior function of an distribution and properties with the distribution, such because the existence of moments as well as the existence of a density function.