Question

Problem 5. Suppose we have finite bivariate population {(x 1​ ,y 1 ),⋯,(x N ,yN )}. We assume N>1. Let τ functionx and τ be the population totals of the x - and y-measurements, respectively; let μ x and μ y be the population means of the x - and y-measurements, respectively; let σ 2 >0 and σ y2 >0 denote the population variances of the x - and y-measurements, respectively. Let σ xy denote the population covariance. Just for reference, σ xy = 1/N ∑i=1n (xi −μx )(yi −μ y ) Suppose {(X1 ,Y1 ),⋯,(Xn ,Y n )}, where n>1, denotes a random sample drawn WITH REPLACEMENT from this population. Let Xˉ andYˉ denote the sample means of the x - and y-measurements in the sample. What is true about the sample {(X 1 ,Y 1 ),⋯,(Xn ,Yn )} ? (a) (X 1 ,Y 1 ) is identically distributed to (X 2​ ,Y 2 ), but they are not independent. (b) (X1 ,Y1 ) is independent of (X 2 ,Y2 ), but the distributions of these two bivariate random variables will be different. (c) Since the pairs (X i ,Yi ) are drawn uniformly at random from the population and since the sample is selected with replacement, the pairs (X 1 ,Y1 ) and (X2 ,Y2 ) are independent and identically distributed bivariate random variables. (d) Since Y 1 is not necessarily independent of X 1 , the bivariate random variable (X 1 ,Y1 ) cannot be independent of the bivariate random variable (X 2 ,Y2 ). (e) Both (a) and (d) are correct. (f) None of the above.

Answer 1

Since the pairs (Xi, Yi) are drawn uniformly at random from the population and the sample is selected with replacement, the pairs (X1, Y1) and (X2, Y2) are independent and identically distributed bivariate random variables.Thus option (c) is correct answer.

Step-by-step explanation:

In this scenario, when a sample is drawn with replacement from a finite population, each pair (Xi, Yi) in the sample is independent as the selection is done uniformly at random with replacement. Consequently, the pairs (X1, Y1) and (X2, Y2) are independent, implying that the joint distribution of (X1, Y1) is identical to that of (X2, Y2).

Drawing with replacement means each draw doesn't affect subsequent ones, ensuring that the probabilities of each outcome remain constant across draws. Therefore, both (X1, Y1) and (X2, Y2) share the same distribution. Additionally, independence between these pairs arises due to the random selection process, leading to no association or correlation between specific pairs in the sample. Hence, they are identically distributed and independent bivariate random variables.

This independence doesn't imply independence between individual variables within each pair. Although (X1, Y1) and (X2, Y2) are independent pairs, within each pair, X1 might not be independent of Y1. However, as joint bivariate random variables, they maintain independence and identical distribution owing to the method of random sampling with replacement from the population.