Question A and B are already answered Please answer quenstio

Question A and B are already answered. Please answer quenstion C( the question about COVARIANCE)

Consider a finite population of numbers x_1, x_2, ..., x_n with x_i {0, 1}, 1 lessthanorequalto i lessthanorequalto N. Write p = 1/N sigma_i=1^N x_i for the population average(numbers of ones). From the population draw two times randomly, without replacement a number. The first draw is called random variable X, the second random variable is called Y. A) Show that E[Y|X] = N_p - X/N - 1. B) Show that E[XY] = p^2 - p(1 - p)/N - 1 C) Consider now multiple draws(without replacement), say 2 lessthanorequalto n lessthanorequalto N. Call still the first draw X and the last draw is called Y(thus the n-th draw) Give an expression for the covariance between X and Y.

Solution

n sampling without replacement, the two sample values X and Y aren\'t independent. Practically, this means that what we got on for the first one X affects what we can get for the second onethat is Y. Mathematically, this means that the covariance between the two X and Y isn\'t zero. That complicates the computations. In particular, if we have a SRS (simple random sample) without replacement, from a population with variance sigma², then the covariance of two namely X and Y of the different sample values is given by

Cov(X,Y) = -Sigma²/N-1 where Nis the total no of population