Question

A populaiton with N units, where each unit has two measurement(x_i,y_i). The population covariance is defined as: $$ S_{xy} =\frac{1}{N-1}\sum_{i=1}^{N}(y_{i}-\bar{y}_{U})(x_{i}-\bar{x}_{U}), \text{where } \bar{y}_{U}=\frac{1}{N}\sum_{i=1}^{N}y_{i} \text{and } \bar{x}_{U}=\frac{1}{N}\sum_{i=1}^{N}x_{i})\,. $$ Choose an SRS of n units out of the N units in the population. The sample covariance is defined as: $$ s_{xy} =\frac{1}{n-1}\sum_{i=1}^{n}(y_{i}-\bar{y})(x_{i}-\bar{x}), \text{where } \bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i} \text{and } \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i})\,. $$ Question: how to prove that: $$ Cov(\bar{x},\bar{y})=\frac{N-n}{Nn}S_{xy}\\ $$

Answer 1