WebCharacterization of a distribution via the moment generating function. The most important property of the mgf is the following. Proposition Let and be two random variables. Denote by and their distribution functions and by and their mgfs. and have the same distribution (i.e., for any ) if and only if they have the same mgfs (i.e., for any ). WebDefinition 3.8.1. The rth moment of a random variable X is given by. E[Xr]. The rth central moment of a random variable X is given by. E[(X − μ)r], where μ = E[X]. Note that the expected value of a random variable is given by the first moment, i.e., when r = 1. Also, the variance of a random variable is given the second central moment.

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