Can this be simplified to below?$\frac{(\sum_{i=1}^{n} X_i Y_i)}{(\sum_{i=1}^{n} X_i^2)} = \frac{\sum_{i=1}^{n} Y_i}{\sum_{i=1}^{n} X_i}$

Question

Can this be simplified to below?

$$\frac{(\sum_{i=1}^{n} X_i Y_i)}{(\sum_{i=1}^{n} X_i^2)} = \frac{\sum_{i=1}^{n} Y_i}{\sum_{i=1}^{n} X_i}$$

Answer 1

No, if the $X_i$ are not all the same.

Answer 2

No. Just try a couple cases with $n=2$. For example, let $X_1=1, X_2=10,Y_1=10,Y_2=1$. Then $$\frac{(\sum_{i=1}^{n} X_i Y_i)}{(\sum_{i=1}^{n} X_i^2)} =\frac{20}{101}\\ \frac{\sum_{i=1}^{n} Y_i}{\sum_{i=1}^{n} X_i}=\frac {11}{11}=1$$