I have been researching about the Cramer-Rao bound and I have found two inequalities:
$$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\theta)\right]^2\right]}$$
$$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{n\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\theta)\right]^2\right]}$$
It is easy to see that the second is more general than the first since:
$$\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\theta)\right]^2\right]}\geq\frac{1}{n\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\theta)\right]^2\right]}$$
I would like to know why there are two bounds. In other words I want to know when I can't use the first one and have to use the second bound.