Problem: Important sum rules for sample moments
Problem
Show that
\[\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }=\sum_{i=1}^{n}{ x_i^2 }-n{\bar{x}}^2\]
Show that
\[\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)\left( y_i-\bar{y} \right) }=\sum_{i=1}^{n}{ x_iy_i }-n\bar{x}\bar{y}\]
Show that if either \(\bar{x}=0\) or \(\bar{y}=0\) then
\[s_{x,y}^2= \frac{1}{n-1}\sum_{i=1}^{n}{ x_iy_i }\]
Show that \(r_{x,y}^2\) and \(s_{x,y}^2\) will always have the same sign
Solution
c.
\[\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)\left( y_i-\bar{y} \right) }=\sum_{i=1}^{n}{ x_iy_i }-n\bar{x}\bar{y}=\sum_{i=1}^{n}{ x_iy_i }\]
