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2. Let $XRV$ with CDF $F_{X}$. A scale functional $?_{X}=T(F_{X})$ is any functional that satisfies (i) $?_{X}?0$ (ii) if $Y=cX$, and $?_{Y}=T(F_{Y})$, then $?_{Y}=?c??_{X}$ for all $c?R$ (iii) if $Y=X+a$, and $?_{Y}=T(F_{Y})$, then $?_{Y}=?_{X}$ for all $c?R$ For instance, the standard deviation $T(F_{X})=Var(X)?$ is a scale functional. Show the following measures are scale functionals. (a) $T(F_{X})=E?X???$, where $?=EX$ (b) For $X$ continuous, $T(F_{X})=2f_{X}(m_{X})1?$, where $f_{X}=F_{X}$, and $m_{X}$ is the median of $X$. [Tip: recall the median is a location functional.]

(a) To show that T(FX)=E|X-?| is a scale functional, we need to show that it satisfies the three conditions:(i) T(FX) is non-negative: E|X-?| is non-n