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Let $X,Y$ be two independent random normal standard distributions. Consider a function $u(x)=\sqrt[ ]{x}$ if $x\geq{}0$ and $u(x)=-2\sqrt[ ]{-x}$ if $x<0$. Define $Z=aX+(1-a)Y$.

Question: How do I find the necessary and sufficient conditions such that $a=\displaystyle\frac{1}{2}$ is a global maximum of $Eu(Z)$.

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Needs to be edited. – Deane Yang May 29 '10 at 15:25
If you meant that $X$ and $Y$ are independent standard normal distributed, then isn't $Eu(Z)$ a fixed function of $a$, which can be evaluated in closed-form? – mr.gondolier May 29 '10 at 17:26

First, $Z$ is distributed like $bX$, with $b>0$ and $b^2=a^2+(1−a)^2$. Second, for every $x$, $u(bx)=\sqrt{b}u(x)$. Third, $E(u(X))=-E(\sqrt{X};X>0)$ is negative. Hence, $E(u(Z))=\sqrt{b}E(u(X))$ is at its maximum when $b$ is at its minimum. This happens when $a=\frac12$.

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