Suppose that $X$ is continuous random variable, taking values in a finite interval $I \subset \mathbb{R}$, with unknown pdf $f$.
Let $x_1, ..., x_n$ be an i.i.d sample of $X$.
Let $\hat{f}$ be a histogram estimator defined as
$$
\hat{f} := (\hat{f_1}, ..., \hat{f_k})
$$
$$
\hat{f} := (n^{-1}\sum_{i=1}^n1\{x_i \in I_1\}, ..., n^{-1}\sum_{i=1}^n1\{x_i \in I_k\})
$$
with $k$ fixed and let $I = \bigcup_{j=1}^kI_j$ be a partition.
(1) Calculate $E(\hat{f}) = (E(\hat{f_1}), ..., E(\hat{f_k}))$.
(2) Which parameters is $\hat{f}$ an unbiased estimator of?
Hint 1: $$E(\hat{f}_1) = E \left[\frac{1}{n} \sum_{i=1}^n 1\{x_i \in I_1\}\right] = \frac{1}{n} \sum_{i=1}^n E\left[1\{x_i \in I_1\}\right]$$ (why are these steps justified?)
Hint 2: Write each $E[1\{x_i \in I_1\}]$ as a probability of a certain event.
Hint 3: The answer to question (2) is essentially what you get in question (1).