The Raw and Central Moments of a Probability Distribution

Moments are expectations of quantities involving a random variable X that reveal characteristics of X’s distribution, most of which deal with its shape.  In the real world, we often can’t justify making assumptions related to an unknown distribution’s shape, which means we can’t enjoy the simplicity and predictability of assuming a bell-shaped curve.  As such, questions about the distributions’s symmetry and tail mass are critical.  We calculate them using the skewness, a function of the third raw moment, and the kurtosis, a function of the fourth raw moment, respectively.  The word document below mathematically derives general formulas for all of a distribution’s moments, provides a mathematical relationship between raw and central moments, and gives formulas to calculate the skewness and kurtosis.

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