I have an estimator, a_hat = (n_1)/(n_0 + n_1), where n_x refers to the frequency of x-valued data. Note that, E(n_0) and E(n_1) exists.

Usually, to show an estimator is unbiased, we need to show that E(a_hat) = a.

However, it can be shown if we do as below

E(a_hat) = E[(n_1)/(n_0 + n_1)] = E(n_0)/[E(n_0) + E(n_1)] = a

However, opting to do as above must have solid reasoning. Can anyone explain how to show the estimator unbiased? Or the reasoning/assumption to use above method? Or links regarding this matter?

Thanks in advance.