1
$\begingroup$

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 $Eu(Z)$.

Improve this question
$\endgroup$
2
  • $\begingroup$ Needs to be edited. $\endgroup$
    – Deane Yang
    Commented May 29, 2010 at 15:25
  • $\begingroup$ 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? $\endgroup$
    – gondolier
    Commented May 29, 2010 at 17:26

1 Answer 1

Reset to default
1
$\begingroup$

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 $a=\frac12$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .