$Var( X_i\epsilon_i)= E[(X_i\epsilon_i)^2]-(E[X_i\epsilon_i])^2 =EX_i^2 \times E\epsilon_i^2-0=(\mu^2+\tau^2)\sigma^2$

##Corrected formula:

$$Var(\frac{U}{V}) \approx (\frac{E[U]}{E[V]})^{2}\cdot(\frac{Var(U)}{(E(U))^2} - \frac{2Cov(U,V)}{E[U]\cdot E[V]} + \frac{Var(V)}{(E[V])^2})$$

and hence in this case(since $EU=0$):

$$Var(\frac{U}{V}) \approx \frac{Var(U)}{(E(V))^2} = \frac{n(\mu^2+\tau^2)\sigma^2}{(n(\mu^2+\tau^2))^2} = \frac{\sigma^2}{n(\mu^2+\tau^2)}$$

$Var( X_i\epsilon_i)= E[(X_i\epsilon_i)^2]-(E[X_i\epsilon_i])^2 =EX_i^2 \times E\epsilon_i^2-0=(\mu^2+\tau^2)\sigma^2$

Corrected formula:

$$Var(\frac{U}{V}) \approx (\frac{E[U]}{E[V]})^{2}\cdot(\frac{Var(U)}{(E(U))^2} - \frac{2Cov(U,V)}{E[U]\cdot E[V]} + \frac{Var(V)}{(E[V])^2})$$

and hence in this case(since $EU=0$):

$$Var(\frac{U}{V}) \approx \frac{Var(U)}{(E(V))^2} = \frac{n(\mu^2+\tau^2)\sigma^2}{(n(\mu^2+\tau^2))^2} = \frac{\sigma^2}{n(\mu^2+\tau^2)}$$

$Var( X_i\epsilon_i)= E[(X_i\epsilon_i)^2]-(E[X_i\epsilon_i])^2 =EX_i^2 \times E\epsilon_i^2-0=(\mu^2+\tau^2)\sigma^2$

Corrected formula:

$$Var(\frac{U}{V}) \approx (\frac{E[U]}{E[V]})^{2}\cdot(\frac{Var(U)}{(E(U))^2} - \frac{2Cov(U,V)}{E[U]\cdot E[V]} + \frac{Var(V)}{(E[V])^2})$$

and hence in this case(since $EU=0$):

$$Var(\frac{U}{V}) \approx \frac{Var(U)}{(E(V))^2} = \frac{n(\mu^2+\tau^2)\sigma^2}{(n(\mu^2+\tau^2))^2} = \frac{\sigma^2}{n(\mu^2+\tau^2)}$$

$Var( X_i\epsilon_i)= E[(X_i\epsilon_i)^2]-(E[X_i\epsilon_i])^2 =EX_i^2 \times E\epsilon_i^2-0=(\mu^2+\tau^2)\sigma^2$

##Corrected formula:

$$Var(\frac{U}{V}) \approx (\frac{E[U]}{E[V]})^{2}\cdot(\frac{Var(U)}{(E(U))^2} - \frac{2Cov(U,V)}{E[U]\cdot E[V]} + \frac{Var(V)}{(E[V])^2})$$

and hence in this case(since $EU=0$):

$$Var(\frac{U}{V}) \approx \frac{Var(U)}{(E(V))^2} = \frac{n(\mu^2+\tau^2)\sigma^2}{(n(\mu^2+\tau^2))^2} = \frac{\sigma^2}{n(\mu^2+\tau^2)}$$