The correlation coefficient of X and Y is
\({\rho _{X,Y}} = \frac{{Cov(X,Y)}}{{{\sigma _X} \times {\sigma _Y}}}\)
Notice first that for such a random sample from the sample distribution,\({F_n}\)the expected values are
\(\begin{aligned}{l}E\left( {{X^*}} \right) &= \bar X\\E\left( {{Y^*}} \right) &= \bar Y\end{aligned}\)
and the variances are given by
\(\begin{aligned}{l}V\left( {{X^*}} \right) &= \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} \\\\V\left( {{Y^*}} \right) &= \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{Y_i} - \bar Y} \right)}^2}} \end{aligned}\)
Finally, to get the sample correlation coefficient of
\(R = \frac{{\sum\limits_{i = 1}^n {\left( {{X_i} - \bar X} \right)} \left( {{Y_i} - \bar Y} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} \sum\limits_{i = 1}^n {{{\left( {{Y_i} - \bar Y} \right)}^2}} } }}\)
it notices that the covariance\(is Cov(X,Y)\)is
\(\begin{aligned}{l}Cov(X,Y) &= E\left( {\left( {{X^*} - \bar X} \right)\left( {{Y^*} - \bar Y} \right)} \right)\\\\ &= \sum\limits_{i = 1}^n {\left( {{X_i} - \bar X} \right)} \left( {{Y_i} - \bar Y} \right)\end{aligned}\)
Because the sample distribution\({F_n}\)is discrete. Finally, by substituting all values and using the fact that\({\sigma _{{X^*}}} = \sqrt {V\left( {{X^*}} \right)} \)it is true that
\(\begin{aligned}{l}{\rho _{X,Y}} &= \frac{{Cov(X,Y)}}{{{\sigma _X} \times {\sigma _Y}}}\\\\ &= \frac{{\sum\limits_{i = 1}^n {\left( {{X_i} - \bar X} \right)} \left( {{Y_i} - \bar Y} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} \sum\limits_{i = 1}^n {{{\left( {{Y_i} - \bar Y} \right)}^2}} } }}\quad \quad \end{aligned}\)
