(a) Let $(X, \mathcal{F}, \nu)$ be a probability space. State the definition of the space $\mathbb{L}^{2}(X, \mathcal{F}, \nu)$. Show that it is a Hilbert space.

(b) Give an example of two real random variables $Z_{1}, Z_{2}$ that are not independent and yet have the same law.

(c) Let $Z_{1}, \ldots, Z_{n}$ be $n$ random variables distributed uniformly on $[0,1]$. Let $\lambda$ be the Lebesgue measure on the interval $[0,1]$, and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Consider the expression

$$D(f):=\operatorname{Var}\left[\frac{1}{n}\left(f\left(Z_{1}\right)+\ldots+f\left(Z_{n}\right)\right)-\int_{[0,1]} f d \lambda\right]$$

where Var denotes the variance and $f \in \mathbb{L}^{2}([0,1], \mathcal{B}, \lambda)$.

Assume that $Z_{1}, \ldots, Z_{n}$ are pairwise independent. Compute $D(f)$ in terms of the variance $\operatorname{Var}(f):=\operatorname{Var}\left(f\left(Z_{1}\right)\right)$.

(d) Now we no longer assume that $Z_{1}, \ldots, Z_{n}$ are pairwise independent. Show that

$$\sup D(f) \geqslant \frac{1}{n},$$

where the supremum ranges over functions $f \in \mathbb{L}^{2}([0,1], \mathcal{B}, \lambda)$ such that $\|f\|_{2}=1$ and $\int_{[0,1]} f d \lambda=0$.

[Hint: you may wish to compute $D\left(f_{p, q}\right)$ for the family of functions $f_{p, q}=\sqrt{\frac{k}{2}}\left(1_{I_{p}}-1_{I_{q}}\right)$ where $1 \leqslant p, q \leqslant k, I_{j}=\left[\frac{j}{k}, \frac{j+1}{k}\right)$ and $1_{A}$ denotes the indicator function of the subset $\left.A .\right]$