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When we speak of moments of a distribution or of a random variable, we are referring to such things as the mean, variance, skewness, and kurtosis.

The first moment of a distribution is its mean. For a discrete random variable y_i, the expectation is given by:

where y_i is the given value of the variable, and p(y_i) is its associated probability. When y_i is a continuous random variable, the expectation is given by:

Notice again that in both cases, whether the variable is discrete or continuous, we are simply summing products of values of the variable with its probability, or density if the variable is continuous. In the case of the discrete variable, the products are “explicit” in that our notation tells us to take each value of y (i.e., y_i) and multiply by the probability of that given value, p(y_i). In the case of a continuous variable, the products are a bit more implicit one might say, since the “probability” of any particular value in a continuous density is equal to 0. Hence, the product y_ip(y_i) is equal to the given value of y_i multiplied by its corresponding density.

The arithmetic mean is a point such that . That is, the sum of deviations around the mean is always equal to 0 for any data set we may consider. In this sense, we say that the arithmetic mean is the center of gravity of a distribution, it is the point that “balances” the distribution (see Figure 2.8).