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Expanding around the expected value $\mu$,because $\mathbb{E}[X-\mu ]=0$. Thereforewith equality exactly when $a=\mu$.

For the absolute loss,Leibniz differentiation gives, with $F(a)={\int }_{-\infty }^{a}f(x)dx$,Thus $H$ decreases while $F(a)<1/2$ and increases while $F(a)>1/2$. It is minimized at any median, characterized in the continuous case by

A function $f$ on an interval is a convex function ifJensen inequality states that for an integrable random variable $X$, when all terms are defined,

The function $\varphi (x)=x\log x$ is convex on $(0,\infty )$ becauseApplying Jensen's inequality to the uniform distribution on the positive numbers $x_1,\dots ,x_n$ givesWriting $S={\sum }_i{x}_i$ and multiplying by $n/S$ yields

The characteristic polynomial of the linear recurrence relation isThe repeated-root rule therefore givesThe condition $x_0=0$ gives $A=0$. The other two conditions giveso $B+C=2$ and $B+2C=3$. Thus $B=C=1$, and

Differentiation under the integral sign first givesOn the other hand,Its integral is zero because $\sin \theta$ vanishes at both endpoints. Hence

Differentiating once more,For $x\ne 0$, the preceding identity and ${\sin }^2\theta +{\cos }^2\theta =1$ giveThe equation extends through $x=0$ in its regular limiting form.