Definition
For a scalar random variable X the characteristic function is defined as the expected value of eitX, where i is the imaginary unit, and t ∈ R is the argument of the characteristic function:
Here FX is the cumulative distribution function of X, and the integral is of the Riemann–Stieltjes kind. If random variable X has a probability density function fX, then the characteristic function is its Fourier transform, and the last formula in parentheses is valid.
It should be noted though, that this convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform. For example some authors define φX(t) = Ee−2πitX, which is essentially a change of parameter. Other notation may be encountered in the literature: as the characteristic function for a probability measure p, or as the characteristic function corresponding to a density f.
Read more about this topic: Characteristic Function (probability Theory)
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