Suppose$X_1, X_2, ... , X_n$ are IID N($\mu$, 1). Prove that the simple estimate $I(X_1\geq0)$ is an unbiased estimate of the parametric function.
To show it is an unbiased estimator, I tried to compute $E(I(X_1\geq0)) = \int_{0}^{\infty}xf(x)dx$, but it diverges. So I have no idea how to prove this is an unbiased estimator (Actually I don't know how to deal with the indicator function here). Can anyone please give me any idea about how I can solve this problem? Thanks a lot!
Hint: That is not the right integral for $E(I(X_1 \ge 0))$. The correct integral is $$\int_{-\infty}^{\infty} I(x\ge 0)f(x)\, dx = \int_0^{\infty}\color{blue}{1}\cdot f(x)\, dx.$$
(This is because $I(x\ge 0) = \begin{cases}\color{blue}{1} &\text{if } x\ge 0 \\ 0 &\text{if } x < 0\end{cases}.$)