I have got the following model: $$Y_i=\beta X_i+\epsilon_i, \hspace{1cm} i=1,...,n$$ where $X_i$ are independent $N(\mu, \tau^2)$ random variables and $\epsilon_i$ are i.i.d. $N(0, \sigma^2)$.
I need to calculate variance of $$\frac{\sum_{i=1}^{n}Y_i}{\sum_{i=1}^{n}X_i}$$ As: $$\frac{\sum_{i=1}^{n}Y_i}{\sum_{i=1}^{n}X_i}=\beta+\frac{\sum_{i=1}^{n}\epsilon_i}{\sum_{i=1}^{n}X_i}$$ I went as follows: $$Var\bigg(\frac{\sum_{i=1}^{n}Y_i}{\sum_{i=1}^{n}X_i}\bigg)=\frac{E[(\sum_{i=1}^{n}\epsilon_i)^2]}{E[(\sum_{i=1}^{n}X_i)^2]}-\frac{E[\sum_{i=1}^{n}\epsilon_i]^2}{E[\sum_{i=1}^{n}X_i]^2}=\frac{E[(\sum_{i=1}^{n}\epsilon_i)^2]}{E[(\sum_{i=1}^{n}X_i)^2]}-0$$ Then: $$\frac{E[(\sum_{i=1}^{n}\epsilon_i)^2]}{E[(\sum_{i=1}^{n}X_i)^2]}=\frac{nE[\epsilon_1^2]}{nE[X_1^2]+n(n-1)E[X_1]^2}=\frac{\sigma^2}{\tau^2+n\mu^2}$$
Am I right?