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[All Steps] Monte Carlo integration [cf. Durr. 2.2.3] Let f:[0,1] \rightarrow R be such that ∫_0^1 f^2(x) d x D_n:=n^-1 ∑_i=1^n f(U_i)-∫_0^1 f(x)

Question: Monte Carlo integration [cf. Durr. 2.2.3] Let $f:[0,1] \rightarrow \mathbb{R}$ be such that $\int_{0}^{1} f^{2}(x) d x<\infty .$ Let $\left(U_{i}\right)$ be i.i.d. Uniform $(0,1)$. Let

\[D_{n}:=n^{-1} \sum_{i=1}^{n} f\left(U_{i}\right)-\int_{0}^{1} f(x) d x\]

Use Chebyshev's inequality to bound $P\left(\left|D_{n}\right|>\varepsilon\right)$.
Show this bound remains true if the $\left(U_{i}\right)$ are only pairwise independent.

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[All Steps] Monte Carlo integration [cf. Durr. 2.2.3] Let f:[0,1] \rightarrow R be such that ∫_0^1 f^2(x) d x D_n:=n^-1 ∑_i=1^n f(U_i)-∫_0^1 f(x)

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