Suppose that $f : [0, 1] → [0, 1]$ and we wish to estimate $$I = \int_{0}^{1} f(x) dx$$
Using the hit-and-miss method, we obtain the estimate
$$\hat I_{HM}=\frac{1}{n}\sum_{i=1}^{n}X_i$$ where $X_1, . . . , X_n$ are an iid sample and $X_i ∼ binom(1, I)$
Using the improved Monte-Carlo method, we obtain the estimate
$$\hat I_{MC}=\frac{1}{n}\sum_{i=1}^{n}f(U_i)$$
where $U_1, . . . , U_n$ are an iid sample of $U(0, 1)$ random variables.
Show that $$Var (X_1) = \int_{0}^{1}f(x) dx − (\int_{0}^{1}f(x) dx)^2$$ and that
$$Var f(U_1) = \int_{0}^{1}f^2(x) dx − (\int_{0}^{1}f(x) dx)^2$$
I don't understand how the structure can be like that .
since $X_1 ∼ binom(1, I)$ I supposed $$Var (X_1)=npq=(1)(I)(1-I)$$
Hint for second part: $E [f(U)^2] = \int_0^1 f(x)^2\; dx$ by the Law of the Unconscious Statistician.