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4.1 Sample Moments

Let x1, x2, x3, . . . . ., xn be a sample of size n from a distribution whose theoretical mean is µ and variance sigma2. This is known as the sample population. The sample mean, xbar is then defined as

Equation 25 (25)

which is just the arithmetic average of the sample. In the limit n -> infty, this can be shown to approach the theoretical mean,

Equation 26 (26)

Similarly, the sample variance, which we denote by s2 is

Equation 27 (27)

which is the average of the squared deviations. In the limit n -> infty, this also approaches the theoretical variance sigma2.

In the case of multivariate samples, for example, (x1, y1), (x2, y2), . . ., the sample means and variances for each variable are calculated as above. In an analogous manner, the sample covariance can be calculated by

Equation 28 (28)

In the limit of infinite n, (28), not surprisingly, also approaches the theoretical covariance (10).

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