being communicated by the notation “E(yi − μ)2.” We can “unpack” this expression to read
where p(yi) is the probability of the given deviation, (yi − μ), for in this case, a discrete random variable.
2.6 MOMENTS AND EXPECTATIONS
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 yi, the expectation is given by:
where yi is the given value of the variable, and p(yi) is its associated probability. When yi 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., yi) and multiply by the probability of that given value, p(yi). 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 yip(yi) is equal to the given value of yi multiplied by its corresponding density.
The arithmetic mean is a point such that
2.6.1 Sample and Population Mean Vectors
We often wish to analyze data simultaneously on several response variables. For this, we require vector and matrix notation to express our responses. The matrix operations presented here are surveyed more comprehensively in the Appendix and in any book on elementary matrix algebra.
Figure 2.8 Because the sum of deviations about the arithmetic mean is always zero, it can be conceptualized as a balance point on a scale.
Consider the following vector:
where y1 is observation 1 up to observation yn.
We can write the sample mean vector
where
The expectation of individual observations within each vector is equal to the population mean μ, of which the expectation of the sample vector y is equal to the population vector, μ. This is simply an extension of scalar algebra to that of matrices:
Likewise, the expectations of individual sample means
We note also that
Recall that we said that the mean is the first moment of a distribution. We discuss the second moment of a distribution, that of the variance, shortly. Before we do so, a brief discussion of estimation is required.
2.7 ESTIMATION AND ESTIMATORS
The goal of statistical inference is, in general, to estimate parameters of a population. We distinguish between point estimators and interval estimators. A point estimator is a function of a sample and is used to estimate a parameter in the population. Because estimates generated by estimators will vary from sample to sample, and thus have a probability distribution associated with them, estimators are also often random variables. For example, the sample mean
More generally, if T is some statistic, then we can use T as an estimator of a population parameter θ. Whether the estimator T is any good depends on
