Question

Characteristic function and moment generating function

Answer 1

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The characteristic function of a uniform U(–1,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued.

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.