The Moments

Moments and Moment Generating Functions
The moments (or raw moments) of a random variable or of a distribution are the expectations of the powers of the random variable which has the given distribution. The first four moments of a random variable or density are used as measures of various .characteristics of the corresponding density. For some of these characteristics,other measures can be defined in terms of quantiles.
The median of a random variable X, denoted by medx, med (X), or (so, is the .5th quantile.
The first moment, locates the" center" of the density of X.
The median of X is also used to indicate a central location of the density of X.
A third measure of location of the density of X, though not necessarily a measure of central location, is the mode of X, which is defined as that point (if such a point exists) at which f (x) attains its maximum.
We previously mentioned that the second moment about the mean, the
variance of a distribution, measures the spread or dispersion of a distribution.
The third moment ?_3 about the mean is sometimes called a measure of asymmetry, or skewness.
The knowledge of the third moment gives almost no clue as to the shape of the distribution, and we mention it mainly to point out that fact.
he fourth moment about the mean is sometimes used as a measure of
excess or kurtosis, which is the degree of flatness of a density near its center. T he fourth moment about the mean is sometimes used as a measure of excess or kurtosis, which is the degree of flatness of a density near its center.