Arithmetic Mean as a Ratio of Moments

In general terms, the ratio of the first moment to the zeroth moment of any distribution defines the arithmetic mean. For an unnormalized number distribution, the zeroth (j = 0) and first (j = 1) moments of the distribution about zero are given, respectively, by [cf. Eq. (P4.2.1)]  

The ratio of these moments is the arithmetic mean of the number distribution and by comparing with Eq.(4.5) we can write 

The arithmetic mean of an unnormalized weight distribution is likewise given by [cf. Eq. (4.10)]  

In the above two examples, we have chosen unnormalized distributions. For normalized distributions, the area under the curve for the differential number distribution [Fig. 4.1(a)] or weight distribution [Fig. 4.2(a)] equals unity. That is, 

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