Generating function of moments

Generating the moment function is a method of defining probability distributions. It is used most often for calculating moments.

Definition

Let there be a random variable with distribution . Then its generating function of moments is called a function that has the form:

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Using the formulas for calculating the expectation, the definition of the generating function of moments can be rewritten in the form:

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that is, the generating function of moments is a two-sided Laplace transform of the distribution of a random variable (up to reflection).

Discrete and absolutely continuous random variables

If a random variable discrete, that is then

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Example. Let be has a Bernoulli distribution. Then

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If a random variable absolutely continuous, meaning it has a density then

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Example. Let be has a standard continuous uniform distribution. Then

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Properties of generating moment functions

The properties of the generating functions of moments are in many respects similar to the properties of the characteristic functions due to the similarity of their definitions.

• The generating function of moments uniquely determines the distribution. Let be are two random variables, and . Then . In particular, if both quantities are absolutely continuous, then the coincidence of the generating functions of the moments implies the coincidence of the densities. If both random variables are discrete, then the coincidence of the generating functions of the moments implies the coincidence of the probability functions.
• The generating function of moments as a function of a random variable is homogeneous:

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• The generating function of moments is the sum of independent random variables equal to the product of their generating functions of moments. Let be are independent random variables. Denote . Then

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Calculation of moments

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