I have some trouble calculating the following variance :
$$V\bigg\{-\frac{2}{\sum_1^n x_i}\bigg\}$$
where $X_i$ follow the Gamma Distribution $\sim \Gamma(2,1/\theta)$.
Attempt :
$X_i$ are independent and then :
$$V[X]=\frac{2}{\theta^2}$$
$$V\bigg\{-\frac{2}{\sum_1^n x_i}\bigg\} = \frac{2^2}{\sum_1^n V(x_i)}= \frac{4}{2n\theta^{-2}}=\frac{2\theta^2}{n}$$
But the correct result should be :
$$V = \frac{\theta^2}{2n}$$
Where is my mistake ?