Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $Eu(Z)$.

share|cite|improve this question
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

1 Answer 1

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 $a=\frac12$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.