The Moment Generating Function (MGF) Theorem provides a powerful tool for characterizing the distribution of a random variable by generating its moments. By taking the expected value of the exponential function of the random variable, the MGF can be used to find all moments of a distribution, as well as to identify the distribution itself when the MGF exists in a neighborhood of zero. This theorem establishes a relationship between the MGF and properties of probability distributions, making it a vital concept in probability theory.
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