Let $X$ be a random variable with mean $\mu$ and variance $\sigma^{2}$. Let

$$G(a)=\mathbb{E}\left[(X-a)^{2}\right]$$

Show that $G(a) \geqslant \sigma^{2}$ for all $a$. For what value of $a$ is there equality?

Let

$$H(a)=\mathbb{E}[|X-a|]$$

Supposing that $X$ has probability density function $f$, express $H(a)$ in terms of $f$. Show that $H$ is minimised when $a$ is such that $\int_{-\infty}^{a} f(x) d x=1 / 2$.