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Assume we have two random variables X and Y. They are joint Gaussian with mean 0 and standard deviation 1 and correlation $\rho$.

  1. What is $E[max\{X^2, Y^2\}]$
  2. What is $E[max\{X,Y\}^2]$

I know that when X and Y are independent, $P\{max\{X,Y\}<m\} = P\{X<m\}*P\{Y<m\}$. Also $X^2$ is a Chi-square random variable. But I am having trouble to use this equation when X and Y are not independent.

saber
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  • Let $Z=X-\rho Y$. Show that $X$ and $Z$ are independent. Write $Y$ as $\frac {X-z} {\rho} $. Work with $(X,Z)$. – Kavi Rama Murthy Nov 25 '21 at 05:07
  • For your purpose, it is easier to use the identity $\max{X^2,Y^2}=\frac12\left[X^2+Y^2+|X^2-Y^2|\right]=\frac12\left[X^2+Y^2+|X+Y||X-Y|\right]$. If the second one is the square of the maximum, then using a similar approach gives the mean and variance of $\max{X,Y}$. – StubbornAtom Nov 25 '21 at 11:27
  • https://math.stackexchange.com/q/2630752/321264 – StubbornAtom Nov 25 '21 at 15:31
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    @StubbornAtom Thank your for the hint! Can you help me with calculating the $E[|X+Y||X-Y|]$? I know X+Y and X-Y are independent. But is |X+Y| and |X-Y| independent as well? Thank you! – saber Nov 25 '21 at 20:10
  • Yes, they are; (measurable) functions of independent random variables are also independent. – StubbornAtom Nov 25 '21 at 20:42

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