Question

Problem 1. Suppose we have a finite population of bivariate data, so the population consists of N pairs (You can think of these as pairs of measurements on a particular object-height and age of an individual, for instance.) Let μΖ and μy be the population means of the z-and y-measurements, respectively, and σ and σ, denote the population variances of the r-and y-measurements. So, for instance, and the corresponding formulas hold for Finally, let and σ ー1 denote the population covariance between r and y, where, as before, μ, and μ, denote the population means of r- and y-measurements. Consider taking a sample of size n, with replacement, from this population. Let (Xi, Y, ), is I, . . . , n denote the (z, y) values for the individuals sampled. Show that the sample covariance n-1 is unbiased for ơzy Hint: Express an expected value as a covariance, then use properties of covariance (how does covariance work with linear transformations?).

Answer 1

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