(a) Let $X$ be a real random variable with $\mathbb{E}\left(X^{2}\right)<\infty$. Show that the variance of $X$ is equal to $\inf _{a \in \mathbb{R}}\left(\mathbb{E}(X-a)^{2}\right)$.

(b) Let $f(x)$ be the indicator function of the interval $[-1,1]$ on the real line. Compute the Fourier transform of $f$.

(c) Show that

$$\int_{0}^{+\infty}\left(\frac{\sin x}{x}\right)^{2} d x=\frac{\pi}{2}$$

(d) Let $X$ be a real random variable and $\widehat{\mu_{X}}$ be its characteristic function.

(i) Assume that $\left|\widehat{\mu_{X}}(u)\right|=1$ for some $u \in \mathbb{R}$. Show that there exists $\theta \in \mathbb{R}$ such that almost surely:

$$u X \in \theta+2 \pi \mathbb{Z}$$

(ii) Assume that $\left|\widehat{\mu_{X}}(u)\right|=\left|\widehat{\mu_{X}}(v)\right|=1$ for some real numbers $u$, $v$ not equal to 0 and such that $u / v$ is irrational. Prove that $X$ is almost surely constant. [Hint: You may wish to consider an independent copy of $X$.]