Paper 2, Section I, F

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 $\int_{-\infty}^{a} f(x) d x=1 / 2$.

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